Encyclopaedia Britannica, 11th Edition, "Dyer, Sir Edward" to "Echidna" Volume 8, Slice 9
VOLUME VIII, SLICE IX
Dyer to Echidna
ARTICLES IN THIS SLICE:
DYER, SIR EDWARD EAST LIVERPOOL DYER, JOHN EAST LONDON DYER, THOMAS HENRY EASTON DYMOKE EAST ORANGE DYNAMICS EASTPORT DYNAMITE EAST PROVIDENCE DYNAMO EAST PRUSSIA DYNAMOMETER EASTWICK, EDWARD BACKHOUSE DYNASTY EATON, DORMAN BRIDGMAN DYSART EATON, MARGARET O'NEILL DYSENTERY EATON, THEOPHILUS DYSPEPSIA EATON, WILLIAM DYSTELEOLOGY EATON, WYATT DZUNGARIA EAU CLAIRE E EAU DE COLOGNE EA EAUX-BONNES EABANI EAVES EACHARD, JOHN EAVESDRIP EADBALD EBBW VALE EADIE, JOHN EBEL, HERMANN WILHELM EADMER EBEL, JOHANN GOTTFRIED EADS, JAMES BUCHANAN EBER, PAUL EAGLE EBERBACH (town of Germany) EAGLEHAWK EBERBACH (monastery of Germany) EAGRE EBERHARD EAKINS, THOMAS EBERHARD, CHRISTIAN AUGUST GOTTLOB EALING EBERHARD, JOHANN AUGUSTUS EAR EBERLIN, JOHANN ERNST EARL EBERS, GEORG MORITZ EARLE, JOHN EBERSWALDE EARLE, RALPH EBERT, FRIEDRICH ADOLF EARL MARSHAL EBINGEN EARLOM, RICHARD EBIONITES EARLSTON EBNER-ESCHENBACH, MARIE EARLY, JUBAL ANDERSON EBOLI EARLY ENGLISH PERIOD EBONY EARN EBRARD, JOHANNES HEINRICH AUGUST EARNEST EBRO EAR-RING EBROIN EARTH EBURACUM EARTH, FIGURE OF THE ECA DE QUEIROZ, JOSE MARIA EARTH CURRENTS ECARTE EARTH-NUT ECBATANA EARTH PILLAR ECCARD, JOHANN EARTHQUAKE ECCELINO DA ROMANO EARTH-STAR ECCENTRIC EARTHWORM ECCHELLENSIS, ABRAHAM EARWIG ECCLES EASEMENT ECCLESFIELD EAST, ALFRED ECCLESHALL EAST ANGLIA ECCLESIA EASTBOURNE ECCLESIASTES EAST CHICAGO ECCLESIASTICAL COMMISSIONERS EASTER ECCLESIASTICAL JURISDICTION EASTER ISLAND ECCLESIASTICAL LAW EASTERN BENGAL AND ASSAM ECCLESIASTICUS EASTERN QUESTION, THE ECGBERT (king of the West Saxons) EAST GRINSTEAD ECGBERT (archbishop of York) EAST HAM ECGFRITH EASTHAMPTON ECGONINE EAST HAMPTON ECHEGARAY Y EIZAGUIRRE, JOSE EAST INDIA COMPANY ECHELON EAST INDIES ECHIDNA EASTLAKE, SIR CHARLES LOCK
DYER, SIR EDWARD (d. 1607), English courtier and poet, son of Sir Thomas Dyer, Kt., was born at Sharpham Park, Somersetshire. He was educated, according to Anthony a Wood, either at Balliol College or at Broadgates Hall, Oxford. He left the university without taking a degree, and after some time spent abroad appeared at Queen Elizabeth's court. His first patron was the earl of Leicester, who seems to have thought of putting him forward as a rival to Sir Christopher Hatton in the queen's favour. He is mentioned by Gabriel Harvey with Sidney as one of the ornaments of the court. Sidney in his will desired that his books should be divided between Fulke Greville (Lord Brooke) and Dyer. He was employed by Elizabeth on a mission (1584) to the Low Countries, and in 1589 was sent to Denmark. In a commission to inquire into manors unjustly alienated from the crown in the west country he did not altogether please the queen, but he received a grant of some forfeited lands in Somerset in 1588. He was knighted and made chancellor of the order of the Garter in 1596. William Oldys says of him that he "would not stoop to fawn," and some of his verses seem to show that the exigencies of life at court oppressed him. He was buried at St Saviour's, Southwark, on the 11th of May 1607. Wood says that many esteemed him to be a Rosicrucian, and that he was a firm believer in alchemy. He had a great reputation as a poet among his contemporaries, but very little of his work has survived. Puttenham in the _Arte of English Poesie_ speaks of "Maister Edward Dyar, for Elegie most sweete, solempne, and of high conceit." One of the poems universally accepted as his is "My Mynde to me a kingdome is." Among the poems in _England's Helicon_ (1600), signed S.E.D., and included in Dr A.B. Grosart's collection of Dyer's works (_Miscellanies of the Fuller Worthies Library_, vol. iv., 1876) is the charming pastoral "My Phillis hath the morninge sunne," but this comes from the _Phillis_ of Thomas Lodge. Grosart also prints a prose tract entitled _The Prayse of Nothing_ (1585). The _Sixe Idillia_ from Theocritus, reckoned by J.P. Collier among Dyer's works, were dedicated to, not written by, him.
DYER, JOHN (c. 1700-1758), British poet, the son of a solicitor, was born in 1699 or 1700 at Aberglasney, in Carmarthenshire. He was sent to Westminster school and was destined for the law, but on his father's death he began to study painting. He wandered about South Wales, sketching and occasionally painting portraits. In 1726 his first poem, _Grongar Hill_, appeared in a miscellany published by Richard Savage, the poet. It was an irregular ode in the so-called Pindaric style, but Dyer entirely rewrote it into a loose measure of four cadences, and printed it separately in 1727. It had an immediate and brilliant success. _Grongar Hill_, as it now stands, is a short poem of only 150 lines, describing in language of much freshness and picturesque charm the view from a hill overlooking the poet's native vale of Towy. A visit to Italy bore fruit in _The Ruins of Rome_ (1740), a descriptive piece in about 600 lines of Miltonic blank verse. He was ordained priest in 1741, and held successively the livings of Calthorp in Leicestershire, Belchford (1751), Coningsby (1752), and Kirby-on-Bane (1756), the last three being Lincolnshire parishes. He married, in 1741, a Miss Ensor, said to be descended from the brother of Shakespeare. In 1757 he published his longest work, the didactic blank-verse epic of _The Fleece_, in four books, discoursing of the tending of sheep, of the shearing and preparation of the wool, of weaving, and of trade in woollen manufactures. The town took no interest in it, and Dodsley facetiously prophesied that "Mr Dyer would be buried in woollen." He died at Coningsby of consumption, on the 15th of December 1758.
His poems were collected by Dodsley in 1770, and by Mr Edward Thomas in 1903 for the _Welsh Library_, vol. iv.
DYER, THOMAS HENRY (1804-1888), English historical and antiquarian writer, was born in London on the 4th of May 1804. He was originally intended for a business career, and for some time acted as clerk in a West India house; but finding his services no longer required after the passing of the Negro Emancipation Act, he decided to devote himself to literature. In 1850 he published the _Life of Calvin_, a conscientious and on the whole impartial work, though the character of Calvin is somewhat harshly drawn, and his influence in the religious world generally is insufficiently appreciated. Dyer's first historical work was the _History of Modern Europe_ (1861-1864; 3rd ed. revised and continued to the end of the 19th century, by A. Hassall, 1901), a meritorious compilation and storehouse of facts, but not very readable. The _History of the City of Rome_ (1865) down to the end of the middle ages was followed by the _History of the Kings of Rome_ (1868), which, upholding against the German school the general credibility of the account of early Roman history, given in Livy and other classical authors, was violently attacked by J.R. Seeley and the _Saturday Review_, as showing ignorance of the comparative method. More favourable opinions of the work were expressed by others, but it is generally agreed that the author's scholarship is defective and that his views are far too conservative. _Roma Regalis_ (1872) and _A Plea for Livy_ (1873) were written in reply to his critics. Dyer frequently visited Greece and Italy, and his topographical works are probably his best; amongst these mention may be made of _Pompeii, its History, Buildings and Antiquities_ (1867, new ed. in Bohn's _Illustrated Library_), and _Ancient Athens, its History, Topography and Remains_ (1873). His last publication was _On Imitative Art_ (1882). He died at Bath on the 30th of January 1888.
DYMOKE, the name of an English family holding the office of king's champion. The functions of the champion were to ride into Westminster Hall at the coronation banquet, and challenge all comers to impugn the king's title (see CHAMPION). The earliest record of the ceremony at the coronation of an English king dates from the accession of Richard II. On this occasion the champion was Sir John Dymoke (d. 1381), who held the manor of Scrivelsby, Lincolnshire, in right of his wife Margaret, granddaughter of Joan Ludlow, who was the daughter and co-heiress of Philip Marmion, last Baron Marmion. The Marmions claimed descent from the lords of Fontenay, hereditary champions of the dukes of Normandy, and held the castle of Tamworth, Leicestershire, and the manor of Scrivelsby, Lincolnshire. The right to the championship was disputed with the Dymoke family by Sir Baldwin de Freville, lord of Tamworth, who was descended from an elder daughter of Philip Marmion. The court of claims eventually decided in favour of the owners of Scrivelsby on the ground that Scrivelsby was held in grand serjeanty, that is, that its tenure was dependent on rendering a special service, in this case the championship.
Sir Thomas Dymoke (1428?-1471) joined a Lancastrian rising in 1469, and, with his brother-in-law Richard, Lord Willoughby and Welles, was beheaded in 1471 by order of Edward IV. after he had been induced to leave sanctuary on a promise of personal safety. The estates were restored to his son Sir Robert Dymoke (d. 1546), champion at the coronations of Richard III., Henry VII. and Henry VIII., who distinguished himself at the siege of Tournai and became treasurer of the kingdom. His descendants acted as champions at successive coronations. Lewis Dymoke (d. 1820) put in an unsuccessful claim before the House of Lords for the barony of Marmion. His nephew Henry (1801-1865) was champion at the coronation of George IV. He was accompanied on that occasion by the duke of Wellington and Lord Howard of Effingham. Henry Dymoke was created a baronet; he was succeeded by his brother John, rector of Scrivelsby (1804-1873), whose son Henry Lionel died without issue in 1875, when the baronetcy became extinct, the estate passing to a collateral branch of the family. After the coronation of George IV. the ceremony was allowed to lapse, but at the coronation of King Edward VII. H.S. Dymoke bore the standard of England in Westminster Abbey.
DYNAMICS (from Gr. [Greek: dynamis], strength), the name of a branch of the science of Mechanics (q.v.). The term was at one time restricted to the treatment of motion as affected by force, being thus opposed to Statics, which investigated equilibrium or conditions of rest. In more recent times the word has been applied comprehensively to the action of force on bodies either at rest or in motion, thus including "dynamics" (now termed kinetics) in the restricted sense and "statics."
ANALYTICAL DYNAMICS.--The fundamental principles of dynamics, and their application to special problems, are explained in the articles MECHANICS and MOTION, LAWS OF, where brief indications are also given of the more general methods of investigating the properties of a dynamical system, independently of the accidents of its particular constitution, which were inaugurated by J.L. Lagrange. These methods, in addition to the unity and breadth which they have introduced into the treatment of pure dynamics, have a peculiar interest in relation to modern physical speculation, which finds itself confronted in various directions with the problem of explaining on dynamical principles the properties of systems whose ultimate mechanism can at present only be vaguely conjectured. In determining the properties of such systems the methods of analytical geometry and of the infinitesimal calculus (or, more generally, of mathematical analysis) are necessarily employed; for this reason the subject has been named Analytical Dynamics. The following article is devoted to an outline of such portions of general dynamical theory as seem to be most important from the physical point of view.
1. _General Equations of Impulsive Motion._
The systems contemplated by Lagrange are composed of discrete particles, or of rigid bodies, in finite number, connected (it may be) in various ways by invariable geometrical relations, the fundamental postulate being that the position of every particle of the system at any time can be completely specified by means of the instantaneous values of a finite number of independent variables q1, q2, ... q_n, each of which admits of continuous variation over a certain range, so that if x, y, z be the Cartesian co-ordinates of any one particle, we have for example
x = [f](q1, q2, ... q_n), y = &c., z = &c., (1)
where the functions [f] differ (of course) from particle to particle. In modern language, the variables q1, q2, ... q_n are _generalized co-ordinates_ serving to specify the _configuration_ of the system; their derivatives with respect to the time are denoted by q`1, q`2, ... q`_n, and are called the _generalized components of velocity_. The continuous sequence of configurations assumed by the system in any actual or imagined motion (subject to the given connexions) is called the _path_.
Impulsive motion.
For the purposes of a connected outline of the whole subject it is convenient to deviate somewhat from the historical order of development, and to begin with the consideration of _impulsive_ motion. Whatever the actual motion of the system at any instant, we may conceive it to be generated instantaneously from rest by the application of proper impulses. On this view we have, if x, y, z be the rectangular co-ordinates of any particle m,
mx` = X', my` = Y', mz` = Z', (2)
where X', Y', Z' are the components of the impulse on m. Now let [delta]x, [delta]y, [delta]z be any infinitesimal variations of x, y, z which are consistent with the connexions of the system, and let us form the equation
[Sigma]m(x`[delta]x + y`[delta]y + z`[delta]z) = [Sigma](X'[delta]x + Y'[delta]y + Z'[delta]z), (3)
where the sign [Sigma] indicates (as throughout this article) a summation extending over all the particles of the system. To transform (3) into an equation involving the variations [delta]q1, [delta]q2, ... of the generalized co-ordinates, we have
dPx dPx x` = ---- q`1 + ---- q`2 + ..., &c., &c. (4) dPq1 dPq2
dPx dPx [delta]x = ---- [delta]q1 + ---- [delta]q2 + ..., &c., &c. (5) dPq1 dPq2
and therefore
[Sigma]m(x`[delta]x + y`[delta]y + z`[delta]z) = (A11q`1 + A12q`2 + ...)[delta]q1 + (A21q`1 + A22q`2 + ...)[delta]q2 + ..., (6)
where _ _ | / dPx \ squared / dPy \ squared / dPz \ squared | \ A_rr = [Sigma]m | ( ----- ) + ( ----- ) + ( ----- ) |, | |_ \dPq_r/ \dPq_r/ \dPq_r/ _| | _ _ > (7) | dPx dPx dPy dPy dPz dPz | | A_rs = [Sigma]m | ----- ----- + ----- ----- + ----- ----- | = A_sr. | |_dPq_r dPq_s dPq_r dPq_s dPq_r dPq_s _| /
If we form the expression for the kinetic energy [Tau] of the system, we find
2[Tau] = [Sigma]m(x` squared + y` squared + z` squared) = A11q`1 squared + A22q`2 squared + ... + 2A12q`1q`2 + ... (8)
The coefficients A11, A22, ... A12, ... are by an obvious analogy called the _coefficients of inertia_ of the system; they are in general functions of the co-ordinates q1, q2, ... . The equation (6) may now be written
dP[Tau] dP[Tau] [Sigma]m(x`[delta]x + y`[delta]y + z`[delta]z) = ------- [delta]q1 + ------- [delta]q2 + ... (9) dPq`1 dPq`2
This maybe regarded as the cardinal formula in Lagrange's method. For the right-hand side of (3) we may write
[Sigma](X'[delta]x + Y'[delta]y + Z'[delta]z) = Q'1[delta]q1 + Q'2[delta]q2 + ... , (10)
where
/ dPx dPy dPz \ Q'_r = [Sigma]( X'----- + Y'----- + Z'----- ). (11) \ dPq_r dPq_r dPq_r/
The quantities Q1, Q2, ... are called the _generalized components of impulse_. Comparing (9) and (10), we have, since the variations [delta]q1, [delta]q2,... are independent,
dP[Tau] dP[Tau] ------- = Q'1, ------- = Q'2, ... (12) dPq`1 dPq`2
These are the general equations of impulsive motion. It is now usual to write
dP[Tau] p_r = ------- (13) dPq`_r
The quantities p1, p2, ... represent the effects of the several component impulses on the system, and are therefore called the _generalized components of momentum_. In terms of them we have
[Sigma]m(x`[delta]x + y`[delta]y + z`[delta]z) = p1[delta]q1 + p2[delta]q2 + ... (14)
Also, since [Tau] is a homogeneous quadratic function of the velocities q`1, q`2 ...,
2[Tau] = p1q`1 + p2q`2 + ... (15)
This follows independently from (14), assuming the special variations [delta]x = x`dt, &c., and therefore [delta]q1 = q`1dt, [delta]q2 = q`2dt, ...
Reciprocal theorems.
Again, if the values of the velocities and the momenta in any other motion of the system through the same configuration be distinguished by accents, we have the identity
p1q`'1 + p2q`'2 + ... = p'1q`1 + p'2q`2 + ..., (16)
each side being equal to the symmetrical expression
A11q`1q''1 + A22q`2q`'2 + ... + A12(q`1q`'2 + q`'1q`2) + ... (17)
The theorem (16) leads to some important reciprocal relations. Thus, let us suppose that the momenta p1, p2, ... all vanish with the exception of p1, and similarly that the momenta p'1, p'2, ... all vanish except p'2. We have then p1q`'1 = p'2q`2, or
q`2 : p1 = q`'1 : p'2 (18)
The interpretation is simplest when the co-ordinates q1, q2 are both of the same kind, e.g. both lines or both angles. We may then conveniently put p1 = p'2, and assert that the velocity of the first type due to an impulse of the second type is equal to the velocity of the second type due to an equal impulse of the first type. As an example, suppose we have a chain of straight links hinged each to the next, extended in a straight line, and free to move. A blow at right angles to the chain, at any point P, will produce a certain velocity at any other point Q; the theorem asserts that an equal velocity will be produced at P by an equal blow at Q. Again, an impulsive couple acting on any link A will produce a certain angular velocity in any other link B; an equal couple applied to B will produce an equal angular velocity in A. Also if an impulse F applied at P produce an angular velocity [omega] in a link A, a couple Fa applied to A will produce a linear velocity [omega]a at P. Historically, we may note that reciprocal relations in dynamics were first recognized by H.L.F. Helmholtz in the domain of acoustics; their use has been greatly extended by Lord Rayleigh.
Velocities in terms of momenta.
The equations (13) determine the momenta p1, p2,... as linear functions of the velocities q`1, q`2,... Solving these, we can express q`1, q`2 ... as linear functions of p1, p2,... The resulting equations give us the velocities produced by any given system of impulses. Further, by substitution in (8), we can express the kinetic energy as a homogeneous quadratic function of the momenta p1, p2,... The kinetic energy, _as so expressed_, will be denoted by [Tau]'; thus
2[Tau]' = A'11p1 squared + A'22p2 squared + ... + 2A'12p - p2 + ... (19)
where A'11, A'22,... A'12,... are certain coefficients depending on the configuration. They have been called by Maxwell the _coefficients of mobility_ of the system. When the form (19) is given, the values of the velocities in terms of the momenta can be expressed in a remarkable form due to Sir W.R. Hamilton. The formula (15) may be written
p1q`1 + p2q`2 + ... = [Tau] + [Tau]', ... (20)
where [Tau] is supposed expressed as in (8), and [Tau]' as in (19). Hence if, for the moment, we denote by [delta] a variation affecting the velocities, and therefore the momenta, but not the configuration, we have
p1[delta]q`1 + q`1[delta]p + p2[delta]q`2 + q`2[delta]p2 + ... = [delta][Tau] + [delta][Tau]'
dP[Tau] dP[Tau] dP[Tau]' dP[Tau]' = ------- [delta]q`1 + ------- [delta]q`2 + ... + -------- [delta]p1 + -------- [delta]p2 + ... (21) dPq`1 dPq`2 dPp1 dPp2
In virtue of (13) this reduces to
dP[Tau]' dP[Tau]' q`1[delta]p1 + q`2[delta]p2 + ... = ------- [delta]p1 + ------- [delta]p2 + ... (22) dPp1 dPp2
Since [delta]p1, [delta]p2, ... may be taken to be independent, we infer that
dP[Tau]' dP[Tau]' q`1 = -------, q`2 = -------, ... (23) dPp1 dPp2
In the very remarkable exposition of the matter given by James Clerk Maxwell in his _Electricity and Magnetism_, the Hamiltonian expressions (23) for the velocities in terms of the impulses are obtained directly from first principles, and the formulae (13) are then deduced by an inversion of the above argument.
Routh's modification.
An important modification of the above process was introduced by E.J. Routh and Lord Kelvin and P.G. Tait. Instead of expressing the kinetic energy in terms of the velocities alone, or in terms of the momenta alone, we may express it in terms of the velocities corresponding to some of the co-ordinates, say q1, q2, ... q_m, and of the momenta corresponding to the remaining co-ordinates, which (for the sake of distinction) we may denote by [chi], [chi]', [chi]", .... Thus, [Tau] being expressed as a homogeneous quadratic function of q`1, q`2, ... q`_m, [chi]`, [chi]`', [chi]`", ..., the momenta corresponding to the co-ordinates [chi], [chi]', [chi]", ... may be written
dP[Tau] dP[Tau] dP[Tau] [kappa] = --------, [kappa]' = ---------, [kappa]" = ------------, ... (24) dP[chi]` dP[chi]`' dP[.[chi]`"
These equations, when written out in full, determine [chi]`, [chi]`', [chi]`", ... as linear functions of q`1, q`2, ... q`_m, [kappa], [kappa]', [kappa]",... We now consider the function
R = [Tau] - [kappa][chi]' - [kappa]'[chi]]`' - [kappa]"[chi]]`" - ..., (25)
supposed expressed, by means of the above relations in terms of q`1, q`2, ... q`_m, [kappa], [kappa]', [kappa]",... Performing the operation [delta] on both sides of (25), we have
dPR dPR dP[Tau] dP[Tau] ----- [delta]q`1 + ... + --------- [delta][kappa] + ... = ------- [delta]q`1 + ... + -------- [delta][chi]` + ... dPq`1 dP[kappa] dPq`1 dP[chi]`
- [kappa]dP[chi]` - [chi]`[delta][kappa] - ... , (26)
where, for brevity, only one term of each type has been exhibited. Omitting the terms which cancel in virtue of (24), we have
dPR dPR dP[Tau] ----- [delta]q`1 + ... + --------- [delta][kappa] + ... = ------- [delta]q`1 + ... - [chi]`[delta][kappa] - ... (27) dPq`1 dP[kappa] dPq`1
Since the variations [delta]q1, [delta]q2, ... [delta]q_m, [delta][kappa], [delta][kappa]', [delta][kappa]", ... may be taken to be independent, we have
dP[Tau] dPR dP[Tau] dPR p1 = ------- = -----, p2 = ------- = -----, ... (28) dPq`1 dPq`1 dPq`2 dPq`2
and
dPR dPR dPR [chi]` = - ---------, [chi]`' = - ----------, [chi]]`" = - ---------, ... (29) dP[kappa] dP[kappa]' dP[kappa]"
An important property of the present transformation is that, when expressed in terms of the new variables, the kinetic energy is the sum of two homogeneous quadratic functions, thus
[Tau] = [@] + K, (30)
where [@] involves the velocities q`1, q`2, ... q`_m alone, and K the momenta [kappa], [kappa]', [kappa]", ... alone. For in virtue of (29) we have, from (25),
/ dPR dPR dPR \ [Tau] = R - ( [kappa] --------- + [kappa]' ---------- + [kappa]" ----------- + ... ), (31) \ dP[kappa] dP[kappa]' dP[kappa]" /
and it is evident that the terms in R which are bilinear in respect of the two sets of variables q`1, q`2, ... q`_m and [kappa], [kappa]', [kappa]", ... will disappear from the right-hand side.
Maximum and minimum energy.
It may be noted that the formula (30) gives immediate proof of two important theorems due to Bertrand and to Lord Kelvin respectively. Let us suppose, in the first place, that the system is started by given impulses of certain types, but is otherwise free. J.L.F. Bertrand's theorem is to the effect that the kinetic energy is _greater_ than if by impulses of the remaining types the system were constrained to take any other course. We may suppose the co-ordinates to be so chosen that the constraint is expressed by the vanishing of the velocities q`1, q`2, ... q`_m, whilst the given impulses are [kappa], [kappa]', [kappa]",... Hence the energy in the actual motion is greater than in the constrained motion by the amount [@].
Again, suppose that the system is started with prescribed velocity components q`1, q`2, ... q`_m, by means of proper impulses of the corresponding types, but is otherwise free, so that in the motion actually generated we have [kappa] = 0, [kappa]' = 0, [kappa]" = 0, ... and therefore K = 0. The kinetic energy is therefore _less_ than in any other motion consistent with the prescribed velocity-conditions by the value which K assumes when [kappa], [kappa]', [kappa]", ... represent the impulses due to the constraints.
Simple illustrations of these theorems are afforded by the chain of straight links already employed. Thus if a point of the chain be held fixed, or if one or more of the joints be made rigid, the energy generated by any given impulses is less than if the chain had possessed its former freedom.
2. _Continuous Motion of a System._
Lagrange's equations.
We may proceed to the continuous motion of a system. The equations of motion of any particle of the system are of the form
mx" = X, my" = Y, mz" = Z (1)
Now let x + [delta]x, y + [delta]y, z + [delta]z be the co-ordinates of m in any arbitrary motion of the system differing infinitely little from the actual motion, and let us form the equation
[Sigma]m(x"[delta]x + y"[delta]y + z"[delta]z) = [Sigma](X[delta]x + Y[delta]y + Z[delta]z) (2)
Lagrange's investigation consists in the transformation of (2) into an equation involving the independent variations [delta]q1, [delta]q2, ... [delta]q_n.
It is important to notice that the symbols [delta] and d/dt are commutative, since
d dx d [delta]x` = --(x + [delta]x) - -- = --[delta]x, &c. (3) dt dt dt
Hence
d [Sigma]m(x"[delta]x + y"[delta]y + z"[delta]z) = -- [Sigma]m(x`[delta]x + y`[delta]y + z`[delta]z) dt - [Sigma]m(x`[delta]x` + y`[delta]y` + z`[delta]z`)
d = --(p1[delta]q1 + p2[delta]q2 + ...) - [delta][Tau], (4) dt
by Sec. 1 (14). The last member may be written
p`1[delta]q1 + p1[delta]q`1 + p`2[delta]q2 + p2[delta]q`2 + ...
dP[Tau] dP[Tau] dP[Tau] dP[Tau] - ------- [delta]q`1 - ------- [delta]q1 - ------- [delta]q`2 - ------- [delta]q2 - ... (5) dPq`1 dPq1 dPq`2 dPq2
Hence, omitting the terms which cancel in virtue of Sec. 1 (13), we find
/ dP[Tau]\ / dP[Tau]\ [Sigma]m(x"[delta]x + y"[delta]y + z"[delta]z) = (p`1 - ------- ) [delta]q1 + (p`2 - ------- ) [delta]q2 + ... (6) \ dPq1 / \ dPq2 /
For the right-hand side of (2) we have
[Sigma](X[delta]x + Y[delta]y + Z[delta]z) = Q1[delta]q1 + Q2[delta]q2 + ..., (7)
/ dPx dPy dPz \ where Q_r = [Sigma]( X ----- + Y ----- + Z ----- ) (8) \ dPq_r dPq_r dPq_r/
The quantities Q1, Q2, ... are called the _generalized components of force_ acting on the system.
Comparing (6) and (7) we find
dP[Tau] dP[Tau] p`1 - ------- = Q1, p`2 - ------- = Q2, ..., (9) dPq`1 dPq`2
or, restoring the values of p1, p2, ...,
d /dP[Tau]\ dP[Tau] d /dP[Tau]\ dP[Tau] -- ( ------- ) - ------- = Q1, -- ( ------- ) - ------- = Q2, ... (10) dt \ dPq`1 / dPq1 dt \ dPq`2 / dPq2
These are Lagrange's general equations of motion. Their number is of course equal to that of the co-ordinates q1, q2, ... to be determined.
Analytically, the above proof is that given by Lagrange, but the terminology employed is of much more recent date, having been first introduced by Lord Kelvin and P.G. Tait; it has greatly promoted the physical application of the subject. Another proof of the equations (10), by direct transformation of co-ordinates, has been given by Hamilton and independently by other writers (see MECHANICS), but the variational method of Lagrange is that which stands in closest relation to the subsequent developments of the subject. The chapter of Maxwell, already referred to, is a most instructive commentary on the subject from the physical point of view, although the proof there attempted of the equations (10) is fallacious.
In a "conservative system" the work which would have to be done by extraneous forces to bring the system from rest in some standard configuration to rest in the configuration (q1, q2, ... q_n) is independent of the path, and may therefore be regarded as a definite function of q1, q2, ... q_n. Denoting this function (the _potential energy_) by V, we have, if there be no extraneous force on the system,
[Sigma](X[delta]x + Y[delta]y + Z[delta]z) = - [delta]V, (11)
and therefore
dPV dPV Q1 = - ----, Q2 = - ----, .... (12) dPq1 dPq2
Hence the typical Lagrange's equation may be now written in the form
d /dP[Tau]\ dP[Tau] dPV -- ( ------- ) - ------- = - -----, (13) dt \dPq`_r / dPq_r dPq_r
or, again,
dP p`_r = - ----- (V - [Tau]) (14) dPq_r
It has been proposed by Helmholtz to give the name _kinetic potential_ to the combination V - [Tau].
As shown under MECHANICS, Sec. 22, we derive from (10)
d[Tau] ------ = Q1q`1 + Q2q`2 + ..., (15) dt
and therefore in the case of a conservative system free from extraneous force,
d --([Tau] + V) = 0 or [Tau] + V = const., (16) dt
which is the equation of energy. For examples of the application of the formula (13) see MECHANICS, Sec. 22.
3. _Constrained Systems._
Case of varying relations.
It has so far been assumed that the geometrical relations, if any, which exist between the various parts of the system are of the type Sec. 1 (1), and so do not contain t explicitly. The extension of Lagrange's equations to the case of "varying relations" of the type
x = f(t, q1, q2,...q_n), y = &c., z = &c., (1)
was made by J.M.L. Vieille. We now have
dPx dPx dPx x` = --- + ---- q`1 + ---- q`2 + ..., &c., &c., (2) dPt dPq1 dPq2
dPx dPx dPx = ---- [delta]q1 + ---- [delta]q2 + ..., &c., &c., (3) dPq1 dPq2
so that the expression Sec. 1 (8) for the kinetic energy is to be replaced by
2[Tau] = [alpha]0 + 2[alpha]1q`1 + 2[alpha]2q`2 + ... + A11q`1 squared + A22q`2 squared + ... + A12q`1q`2 + ..., (4)
where _ _ \ | /dPx\ squared /dPy\ squared /dPz\ squared | | a0 = [Sigma]m |( --- ) + ( --- ) + ( --- ) |, | |_\dPt/ \dPt/ \dPt/ _| | _ _ > (5) | dPx dPx dPy dPy dPz dPz | | a_r = [Sigma]m | --- ----- + --- ----- + --- ----- |, | |_dPt dPq_r dPt dPq_r dPt dPq_r_| | /
and the forms of A_rr, A_rs are as given by Sec. 1 (7). It is to be remembered that the coefficients [alpha]0, [alpha]1, [alpha]2, ... A11, A22, ... A12 ... will in general involve t explicitly as well as implicitly through the co-ordinates q1, q2,... Again, we find
[Sigma]m(x`[delta]x + y`[delta]y + z`[delta]z) =
([alpha]1 + A11q`1 + A12q`2 + ...)[delta]q1 + ([alpha]2 + A21q`1 + A22q`2 + ...)dPq2 + ...
dP[Tau] dP[Tau] = ------- [delta]q1 + ------- [delta]q2 + ... dPq`1 dPq`2
= p1[delta]q1 + p2[delta]q2 + ..., (6)
where p_r is defined as in Sec. 1 (13). The derivation of Lagrange's equations then follows exactly as before. It is to be noted that the equation Sec. 2 (15) does not as a rule now hold. The proof involved the assumption that [Tau] is a homogeneous quadratic function of the velocities q`1, q`2....
It has been pointed out by R.B. Hayward that Vieille's case can be brought under Lagrange's by introducing a new co-ordinate ([chi]) in place of t, so far as it appears explicitly in the relations (1). We have then
2[Tau] = [alpha]0[chi]` squared + 2([alpha]1q`1 + [alpha]2q`2 + ...)[chi]` + A11q`1 squared + A22q`2 squared + ... + 2A12q`1q`2 + .... (7)
The equations of motion will be as in Sec. 2 (10), with the additional equation
d dP[Tau] dP[Tau] -- -------- - ------- = X, (8) dt dP[chi]` dP[chi]
where X is the force corresponding to the co-ordinate [chi]. We may suppose X to be adjusted so as to make [chi]" = 0, and in the remaining equations nothing is altered if we write t for [chi] before, instead of after, the differentiations. The reason why the equation Sec. 2 (15) no longer holds is that we should require to add a term X[chi]` on the right-hand side; this represents the rate at which work is being done by the constraining forces required to keep [chi]` constant.
As an example, let x, y, z be the co-ordinates of a particle relative to axes fixed in a solid which is free to rotate about the axis of z. If [phi] be the angular co-ordinate of the solid, we find without difficulty
2[Tau] = m(x` squared + y` squared +z` squared) + 2[phi]`m(xy` - yx`) + {I + m(x squared + y squared)}[phi]` squared, (9)
where I is the moment of inertia of the solid. The equations of motion, viz.
d dP[Tau] dP[Tau] d dP[Tau] dP[Tau] d dP[Tau] dP[Tau] -- ------ - ------ = X, -- ------- - ------- = Y, -- ------- - ------- = Z, (10) dt dPx` dPx dt dPy` dPy dt dPz` dPz
d dP[Tau] dP[Tau] and -- -------- - ------- = [Phi], (11) dt dP[phi]` dP[phi]
become
m(x" - 2[phi]`y` - x[phi]` squared - y[phi]") = X, m(y" + 2[phi]`x` - y[phi]` squared + x[phi]`) = Y, mz" = Z, (12) _ _ d | / \ | and -- |(I + m(x squared + y squared)) [phi]` + m(xy` - yx`)| = [Phi]. (13) dt |_\ / _|
If we suppose [Phi] adjusted so as to maintain [phi]" = 0, or (again) if we suppose the moment of inertia I to be infinitely great, we obtain the familiar equations of motion relative to moving axes, viz.
m(x" - 2[omega]y` - [omega] squaredx) = X, m(y" + 2[omega]x` - [omega] squaredy) = Y, mz" = Z, (14)
where [omega] has been written for [phi]. These are the equations which we should have obtained by applying Lagrange's rule at once to the formula
2[Tau] = m(x` squared + y` squared + z` squared) + 2m[omega](xy` - yx`) + m[omega] squared(x squared + y squared), (15)
which gives the kinetic energy of the particle referred to axes rotating with the constant angular velocity [omega]. (See MECHANICS, Sec. 13.)
More generally, let us suppose that we have a certain group of co-ordinates [chi], [chi]', [chi]", ... whose absolute values do not affect the expression for the kinetic energy, and that by suitable forces of the corresponding types the velocity-components [chi]`, [chi]`', [chi]`", ... are maintained constant. The remaining co-ordinates being denoted by q1, q2, ... q_n, we may write
2[Tau] = [@] + [Tau]0 + 2([alpha]1q`1 + [alpha]2q`2 + ...)[chi]` + 2([alpha]'1q`1 + [alpha]'2q`2 + ...)[chi]`' + ..., (16)
where [@] is a homogeneous quadratic function of the velocities q`1, q`2, ... q`_n of the type Sec.1 (8), whilst [Tau]0 is a homogeneous quadratic function of the velocities [chi]`,[chi]`', [chi]`", ... alone. The remaining terms, which are bilinear in respect of the two sets of velocities, are indicated more fully. The formulae (10) of Sec. 2 give n equations of the type
d /dP[@]\ /dP[@]\ dP[Tau]0 --( ----- ) - ( ----- ) + (r, 1)q`1 + (r, 2)q`2 + ... - -------- = Q_r (17) dt \dPq_r/ \dPq_r/ dPq_r
where
/dPa_r dPa_s\ /dPa'_r dPa'_s\ (r, s) = ( ----- - ----- )[chi]` + ( ------ - ------ )[chi]`' + .... (18) \dPq_s dPq_r/ \dPq_s dPq_r/
These quantities (r, s) are subject to the relations
(r, s) = -(s, r), (r, r) = 0 (19)
The remaining dynamical equations, equal in number to the co-ordinates [chi], [chi]', [chi]", ..., yield expressions for the forces which must be applied in order to maintain the velocities [chi]`, [chi]`', [chi]`", ... constant; they need not be written down. If we follow the method by which the equation of energy was established in Sec. 2, the equations (17) lead, on taking account of the relations (19), to
d --([@] - [Tau]0) = Q1q`1 + Q2q`2 + ... + Q_nq`_n, (20) dt
or, in case the forces Q_r depend only on the co-ordinates q1, q2, ... q_n and are conservative,
[@] + V - [Tau]0 = const. (21)
The conditions that the equations (17) should be satisfied by zero values of the velocities q`1, q`2, ... q`_n are
dP[Tau]0 Q_r = - --------, (22) dPq_r
or in the case of conservative forces
dP ------ (V - [Tau]0) = 0, (23) dPq_r
i.e. the value of V - [Tau]0 must be _stationary_.
Rotating axes.
We may apply this to the case of a system whose configuration relative to axes rotating with constant angular velocity ([omega]) is defined by means of the n co-ordinates q1, q2, ... q_n. This is important on account of its bearing on the kinetic theory of the tides. Since the Cartesian co-ordinates x, y, z of any particle m of the system relative to the moving axes are functions of q1, q2, ... q_n, of the form Sec. 1 (1), we have, by (15)
2[@] = [Sigma]m(x` squared + y` squared + z` squared), 2[Tau]0 = [omega] squared[Sigma]m(x squared + y squared), (24)
/ dPy dPx \ a_r= [Sigma]m( x----- - y----- ), (25) \ dPq_r dPq_r/
whence
dP(x, y) (r, s) = 2[omega].[Sigma]m ------------. (26) dP(q_s, q_r)
The conditions of relative equilibrium are given by (23).
It will be noticed that this expression V - [Tau]0, which is to be stationary, differs from the true potential energy by a term which represents the potential energy of the system in relation to fictitious "centrifugal forces." The question of stability of relative equilibrium will be noticed later (Sec. 6).
It should be observed that the remarkable formula (20) may in the present case be obtained directly as follows. From (15) and (14) we find
d[Tau] d ------ = --([@] + [Tau]0) + [omega].[Sigma]m(xy" - yx") dt dt
d = --([@] - [Tau]0) + [omega].[Sigma](xY - yX). (27) dt
This must be equal to the rate at which the forces acting on the system do work, viz. to
[omega][Sigma](xY - yX) + Q1q`1 + Q2q`2 + ... + Q_nq`_n,
where the first term represents the work done in virtue of the rotation.
Constrained systems.
We have still to notice the modifications which Lagrange's equations undergo when the co-ordinates q1, q2, ... q_n are not all independently variable. In the first place, we may suppose them connected by a number m ( < n) of relations of the type
A(t, q1, q2, ... q_n) = 0, B(t, q1, q2, ... q_n) = 0, &c. (28)
These may be interpreted as introducing partial constraints into a previously free system. The variations [delta]q1, [delta]q2, ... [delta]q_n in the expressions (6) and (7) of Sec. 2 which are to be equated are no longer independent, but are subject to the relations
dPA dPA dPB dPB ---- [delta]q1 + ---- [delta]q2 + ... = 0, ---- [delta]q1 + ---- [delta]q2 + ... = 0, &c. (29) dPq1 dPq2 dPq1 dPq2
Introducing indeterminate multipliers [lambda], mu, ..., one for each of these equations, we obtain in the usual manner n equations of the type
d dP[Tau] dP[Tau] dPA dPB -- ------- - ------- = Q_r + [lambda] ----- + mu ----- + ..., (30) dt dPq`_r dPq_r dPq_r dPq_r
in place of Sec. 2 (10). These equations, together with (28), serve to determine the n co-ordinates q1, q2, ... q_n and the m multipliers [lambda], mu, ....
When t does not occur explicitly in the relations (28) the system is said to be _holonomic_. The term connotes the existence of integral (as opposed to differential) relations between the co-ordinates, independent of the time.
Again, it may happen that although there are no prescribed relations between the co-ordinates q1, q2, ... q_n, yet from the circumstances of the problem certain geometrical conditions are imposed on their _variations_, thus
A1[delta]q1 + A2[delta]q2 + ... = 0, B1[delta]q1 + B2[delta]q2 + ... = 0, &c., (31)
where the coefficients are functions of q1, q2, ... q_n and (possibly) of t. It is assumed that these equations are not integrable as regards the variables q1, q2, ... q_n; otherwise, we fall back on the previous conditions. Cases of the present type arise, for instance, in ordinary dynamics when we have a solid rolling on a (fixed or moving) surface. The six co-ordinates which serve to specify the position of the solid at any instant are not subject to any necessary relation, but the conditions to be satisfied at the point of contact impose three conditions of the form (31). The general equations of motion are obtained, as before, by the method of indeterminate multipliers, thus
d dP[Tau] dP[Tau] -- ------- - ------- = Q_r + [lambda]A_r + muB_r + ... (32) dt dPq`_r dPq_r
The co-ordinates q1, q2, ... q_n, and the indeterminate multipliers [lambda], mu, ..., are determined by these equations and by the velocity-conditions corresponding to (31). When t does not appear explicitly in the coefficients, these velocity-conditions take the forms
A1q`1 + A2q`2 + ... = 0, B1q`1 + B2q`2 + ... = 0, &c. (33)
Systems of this kind, where the relations (31) are not integrable, are called _non-holonomic_.
4. _Hamiltonian Equations of Motion._
In the Hamiltonian form of the equations of motion of a conservative system with unvarying relations, the kinetic energy is supposed expressed in terms of the _momenta_ p1, p2, ... and the co-ordinates q1, q2, ..., as in Sec. 1 (19). Since the symbol [delta] now denotes a variation extending to the co-ordinates as well as to the momenta, we must add to the last member of Sec. 1 (21) terms of the types
dP[Tau] dP[Tau]' ------- [delta]q1 + -------- [delta]q1 + ... (1) dPq1 dPq1
Since the variations [delta]p1, [delta]p2, ... [delta]q1, [delta]q2, ... may be taken to be independent, we infer the equations Sec. 1 (23) as before, together with
dP[Tau] dP[Tau]' dP[Tau] dP[Tau]' ------ = - --------, ------- = - --------, ..., (2) dPq1 dPq1 dPq2 dPq2
Hence the Lagrangian equations Sec. 2 (14) transform into
dP dP p`1 = - ----([Tau]' + V), p`2 = ---- ([Tau]' + V), ... (3) dPq1 dPq2
If we write
H = [Tau]' + V, (4)
so that H denotes the _total energy_ of the system, supposed expressed in terms of the new variables, we get
dPH dPH p`1 = - ----, p`2 = - ----, ... (5) dPq1 dPq2
If to these we join the equations
dPH dPH q`1 = ----, q`2 = ----, ..., (6) dPp1 dPp2
which follow at once from Sec. 1 (23), since V does not involve p1, p2, ..., we obtain a complete system of differential equations _of the first order_ for the determination of the motion.
The equation of energy is verified immediately by (5) and (6), since these make
dH dPH dPH dPH dPH -- = ---- p`1 + ---- p`2 + ... + ---- q`1 + ---- q`2 + ... = 0. (7) dt dPp1 dPp2 dPq1 dPq2
The Hamiltonian transformation is extended to the case of varying relations as follows. Instead of (4) we write
H = p1q`1 + p2q`2 + ... - [Tau] + V, (8)
and imagine H to be expressed in terms of the momenta p1, p2, ..., the co-ordinates q1, q2, ..., and the time. The internal forces of the system are assumed to be conservative, with the potential energy V. Performing the variation [delta] on both sides, we find
dP[Tau] dPV [delta]H = q`1[delta]p1 + ... - ------- [delta]q1 + ---- [delta]q + ..., (9) dPq1 dPq1
terms which cancel in virtue of the definition of p1, p2, ... being omitted. Since [delta]p1, [delta]p2, ..., [delta]q1, [delta]q2, ... may be taken to be independent, we infer
dPH dPH q`1 = ----, q`2 = ----, ..., (10) dPp1 dPp2
and
dP dPH dP dPH ---- ([Tau] - V) = - ----, ----([Tau] - V) = - ----, .... (11) dPq1 dPq1 dPq2 dPq2
It follows from (11) that
dPH dPH p`1 = - ----, p`2 = - ----, .... (12) dPq1 dPq2
The equations (10) and (12) have the same form as above, but H is no longer equal to the energy of the system.
5. _Cyclic Systems._
A _cyclic_ or _gyrostatic_ system is characterized by the following properties. In the first place, the kinetic energy is not affected if we alter the absolute values of certain of the co-ordinates, which we will denote by [chi], [chi]', [chi]", ..., provided the remaining co-ordinates q1, q2, ... q_m and the velocities, including of course the velocities [.[chi]], [.[chi]]', [.[chi]]", ..., are unaltered. Secondly, there are no forces acting on the system of the types [chi], [chi]', [chi]", .... This case arises, for example, when the system includes gyrostats which are free to rotate about their axes, the co-ordinates [chi], [chi]', [chi]", ... then being the angular co-ordinates of the gyrostats relatively to their frames. Again, in theoretical hydrodynamics we have the problem of moving solids in a frictionless liquid; the ignored co-ordinates [chi], [chi]', [chi]", ... then refer to the fluid, and are infinite in number. The same question presents itself in various physical speculations where certain phenomena are ascribed to the existence of _latent motions_ in the ultimate constituents of matter. The general theory of such systems has been treated by E.J. Routh, Lord Kelvin, and H.L.F. Helmholtz.
Routh's equations.
If we suppose the kinetic energy [Tau] to be expressed, as in Lagrange's method, in terms of the co-ordinates and the velocities, the equations of motion corresponding to [chi], [chi]', [chi]'', ... reduce, in virtue of the above hypotheses, to the forms
d dP[Tau] d dP[Tau] d dP[Tau] -- --------- = 0, -- --------- = 0, -- --------- = 0, ..., (1) dt dP[chi]` dt dP[chi]`' dt dP[chi]`"
whence
dP[Tau] dP[Tau] dP[Tau] -------- = [kappa], --------- = [kappa]', --------- = [kappa]", ..., (2) dP[chi]` dP[chi]`' dP[chi]`"
where [kappa], [kappa]', [kappa]", ... are the constant momenta corresponding to the cyclic co-ordinates [chi], [chi]', [chi]", .... These equations are linear in [.[chi]], [.[chi]]', [.[chi]]", ...; solving them with respect to these quantities and substituting in the remaining Lagrangian equations, we obtain m differential equations to determine the remaining co-ordinates q1, q2, ... q_m. The object of the present investigation is to ascertain the general form of the resulting equations. The retained co-ordinates q1, q2, ... q_m may be called (for distinction) the _palpable_ co-ordinates of the system; in many practical questions they are the only co-ordinates directly in evidence.
If, as in Sec. 1 (25), we write
R = [Tau] - [kappa][chi]` - [kappa]'[chi]`' - [kappa]"[chi]`" - ..., (3)
and imagine R to be expressed by means of (2) as a quadratic function of q`1, q`2, ... q`_m, [kappa], [kappa]', [kappa]", ... with coefficients which are in general functions of the co-ordinates q1, q2, ... q_m, then, performing the operation [delta] on both sides, we find
dPR dPR dPR dP[Tau] dP[Tau] -----[delta]q`1 + ... + ---------[delta][kappa] + ... + ----[delta]q1 + ... = -------[delta]q`1 + ... + -------[delta]q1 + ... dPq`1 dP[kappa] dPq1 dPq`1 dPq1
dP[Tau] dP[Tau] + --------[delta][chi]` + ... + --------[delta]q1 + ... - [kappa][delta][chi]` - [chi]`[delta][kappa] - .... (4) dP[chi]` dP[chi]1
Omitting the terms which cancel by (2), we find
dP[Tau] dPR dP[Tau] dPR ------- = -----, ------- = -----, ..., (5) dPq`1 dPq`1 dPq`2 dPq`2
dP[Tau] dPR dP[Tau] dPR ------- = ----, ------- = ----, ..., (6) dPq1 dPq1 dPq2 dPq2
dPR dPR dPR [chi]` = - ---------, [chi]`' = - ----------, [chi]`" = - ----------, ... (7) dP[kappa] dP[kappa]' dP[kappa]"
Substituting in Sec. 2 (10), we have
d dPR dPR d dPR dPR -- ----- - ----- = Q1, -- ----- - ---- = Q2, ... (8) dt dPq`1 dPq1 dt dPq`2 dPq2
These are Routh's forms of the modified Lagrangian equations. Equivalent forms were obtained independently by Helmholtz at a later date.
Kelvin's equations.
The function R is made up of three parts, thus
R = R(2,0) + R(1,1) + R(0,2), ... (9)
where R(2,0) is a homogeneous quadratic function of q`1, q`2, ... q`_m, R(0,2) is a homogeneous quadratic function of [kappa], [kappa]', [kappa]", ..., whilst R(1,1) consists of products of the velocities q`1, q`2, ... q`_m into the momenta [kappa], [kappa]', [kappa]".... Hence from (3) and (7) we have
/ dPR dPR dPR \ [Tau] = R - ( [kappa] --------- + [kappa]'---------- + [kappa]" ---------- + ...) \ dP[kappa] dP[kappa]' dP[kappa]" /
= R(2,0) - R(0,2). (10)
If, as in Sec. 1 (30), we write this in the form
[Tau] = [@] + [Kappa], (11)
then (3) may be written
R = [@] - [Kappa] + ss1q`1 + ss2q`2 + ..., (12)
where ss1, ss2, ... are linear functions of [kappa], [kappa]', [kappa]", ..., say
ss_r = [alpha]_r[kappa] + [alpha]'_r[kappa]' + [alpha]"_r[kappa]" + ..., (13)
the coefficients [alpha]_r, [alpha]'_r, [alpha]"_r, ... being in general functions of the co-ordinates q1, q2, ... q_m. Evidently ss_r denotes that part of the momentum-component dPR/dPq`_r which is due to the cyclic motions. Now
d dPR d / dP[@] \ d dP[@] dPss_r dPss_r -- ------ = -- ( ------ + ss_r) = -- ------ + -----q`1 + -----q`2 + ..., (14) dt dPq`_r dt \dPq`_r / dt dPq`_r dPq1 dPq2
dPR dP[@] dP[Kappa] dPss1 dPss2 ----- = ----- - --------- + -----q`1 + -----q`2 + .... (15) dPq_r dPq_r dPq_r dPq_r dPq_r
Hence, substituting in (8), we obtain the typical equation of motion of a gyrostatic system in the form
d dP[@] dP[@] dP[Kappa] -- ------ - ----- + (r, 1)q`1 + (r, 2)q`2 + ... + (r, s)q`_s + ... + --------- = Q_r, (16) dt dPq`_r dPq_r dPq_r
where
dPss_r dPss_s (r, s) = ----- - -----. (17) dPq_s dPq_r
This form is due to Lord Kelvin. When q1, q2, ... q_m have been determined, as functions of the time, the velocities corresponding to the cyclic co-ordinates can be found, if required, from the relations (7), which may be written
dP[Kappa] \ [Chi]` = --------- - [alpha]1q`1 - [alpha]2q`2 - ..., | dP[kappa] | | dP[Kappa] > (18) [Chi]`' = ---------- - [alpha]'1q`1 - [alpha]'2q`2 - ..., | dP[kappa]' | | &c., &c. /
It is to be particularly noticed that
(r, r) = 0, (r, s) = -(s, r). (19)
Hence, if in (16) we put r = 1, 2, 3, ... m, and multiply by q`1, q`2, ... q`_m respectively, and add, we find
d --([@] + [Kappa]) = Q1q`1 + Q2q`2 + ..., (20) dt
or, in the case of a conservative system
[@] + V + [Kappa] = const., (21)
which is the equation of energy.
The equation (16) includes Sec. 3 (17) as a particular case, the eliminated co-ordinate being the angular co-ordinate of a rotating solid having an infinite moment of inertia.
In the particular case where the cyclic momenta [kappa], [kappa]', [kappa]", ... are all zero, (16) reduces to
d dP[@] dP[@] -- ------ - ----- = Q_r. (22) dt dPq`_r dPq_r
The form is the same as in Sec. 2, and the system now behaves, as regards the co-ordinates q1, q2, ... q_m, exactly like the acyclic type there contemplated. These co-ordinates do not, however, now fix the position of every particle of the system. For example, if by suitable forces the system be brought back to its initial configuration (so far as this is defined by q1, q2, ..., q_m), after performing any evolutions, the ignored co-ordinates [chi], [chi]', [chi]", ... will not in general return to their original values.
If in Lagrange's equations Sec. 2 (10) we reverse the sign of the time-element dt, the equations are unaltered. The motion is therefore reversible; that is to say, if as the system is passing through any configuration its velocities q`1, q`2, ..., q`_m be all reversed, it will (if the forces be the same in the same configuration) retrace its former path. But it is important to observe that the statement does not in general hold of a gyrostatic system; the terms of (16), which are linear in q`1, q`2, ..., q`_m, change sign with dt, whilst the others do not. Hence the motion of a gyrostatic system is not reversible, unless indeed we reverse the cyclic motions as well as the velocities q`1, q`2, ..., q`_m. For instance, the precessional motion of a top cannot be reversed unless we reverse the spin.
Kinetostatics.
The _conditions of equilibrium_ of a system with latent cyclic motions are obtained by putting q`1 = 0, q`2 = 0, ... q`_m = 0 in (16); viz. they are
dP[Kappa] dP[Kappa] Q1 = ---------, Q2 = ---------, ... (23) dPq1 dPq2
These may of course be obtained independently. Thus if the system be guided from (apparent) rest in the configuration (q1, q2, ... q_m) to rest in the configuration q1 + [delta]q1, q2 + [delta]q2, ..., q_m + [delta]q_m, the work done by the forces must be equal to the increment of the kinetic energy. Hence
Q1[delta]q1 + Q2[delta]q2 + ... = [delta][Kappa], (24)
which is equivalent to (23). The conditions are the same as for the equilibrium of a system without latent motion, but endowed with potential energy [Kappa]. This is important from a physical point of view, as showing how energy which is apparently potential may in its ultimate essence be kinetic.
By means of the formulae (18), which now reduce to
dP[Kappa] dP[Kappa] dP[Kappa] [chi]` = ---------, [chi]`' = ----------, [chi]`" = ---------- ..., (25) dP[kappa] dP[kappa]' dP[kappa]"
[Kappa] may also be expressed as a homogeneous quadratic function of the cyclic velocities [.[chi]], [.[chi]]', [.[chi]]", ... Denoting it in this form by [Tau]0, we have
[delta]([Tau]0 + [Kappa] = 2[delta][Kappa] = [delta]([kappa] [chi]` + [kappa]'[chi]`' + [kappa]"[chi]`" + ...). (26)
Performing the variations, and omitting the terms which cancel by (2) and (25), we find
dP[Tau]0 dP[Kappa] dP[Tau]0 dP[Kappa] -------- = - ---------, -------- = - ---------, ..., (27) dPq1 dPq1 dPq2 dPq2
so that the formulae (23) become
dP[Tau]0 dP[Tau]0 Q1 = - --------, Q2 = - --------, ... (28) dPq1 dPq2
A simple example is furnished by the top (MECHANICS, Sec. 22). The cyclic co-ordinates being [psi], [phi], we find
( mu - [nu] cos [theta]) squared [nu] squared 2[@] = A[theta]` squared, 2[Kappa] = ----------------------- + -----, A sin squared [theta] C
2[Tau]0 = A sin squared[theta][psi]` squared + C([phi]` + [psi] cos [theta]) squared, (29)
whence we may verify that dP[Tau]0/dP[theta] = - dP[Kappa]/dP[theta] in accordance with (27). And the condition of equilibrium
dP[Kappa] dPV --------- = - --------- (30) dP[theta] dP[theta]
gives the condition of steady precession.
6. _Stability of Steady Motion._
The small oscillations of a conservative system about a configuration of equilibrium, and the criterion of stability, are discussed in MECHANICS, Sec. 23. The question of the stability of given types of motion is more difficult, owing to the want of a sufficiently general, and at the same time precise, definition of what we mean by "stability." A number of definitions which have been propounded by different writers are examined by F. Klein and A. Sommerfeld in their work _Ueber die Theorie des Kreisels_ (1897-1903). Rejecting previous definitions, they base their criterion of stability on the character of the changes produced in the _path_ of the system by small arbitrary disturbing impulses. If the undisturbed path be the _limiting form_ of the disturbed path when the impulses are indefinitely diminished, it is said to be stable, but not otherwise. For instance, the vertical fall of a particle under gravity is reckoned as stable, although for a _given_ impulsive disturbance, however small, the deviation of the particle's position at any time t from the position which it would have occupied in the original motion increases indefinitely with t. Even this criterion, as the writers quoted themselves recognize, is not free from ambiguity unless the phrase "limiting form," as applied to a path, be strictly defined. It appears, moreover, that a definition which is analytically precise may not in all cases be easy to reconcile with geometrical prepossessions. Thus a particle moving in a circle about a centre of force varying inversely as the cube of the distance will if slightly disturbed either fall into the centre, or recede to infinity, after describing in either case a spiral with an infinite number of convolutions. Each of these spirals has, analytically, the circle as its limiting form, although the motion in the circle is most naturally described as unstable.
A special form of the problem, of great interest, presents itself in the steady motion of a gyrostatic system, when the non-eliminated co-ordinates q1, q2, ... q_m all vanish (see Sec. 5). This has been discussed by Routh, Lord Kelvin and Tait, and Poincare. These writers treat the question, by an extension of Lagrange's method, as a problem of small oscillations. Whether we adopt the notion of stability which this implies, or take up the position of Klein and Sommerfeld, there is no difficulty in showing that stability is ensured if V + [Kappa] be a minimum as regards variations of q1, q2, ... q_m. The proof is the same as that of Dirichlet for the case of statical stability.
We can illustrate this condition from the case of the top, where, in our previous notation,
( mu - [nu]cos [theta]) squared [nu] squared V + [Kappa] = Mgh cos[theta] + ---------------------- + -----. (1) 2A sin squared [theta] 2C
To examine whether the steady motion with the centre of gravity vertically above the pivot is stable, we must put mu = [nu]. We then find without difficulty that V + [Kappa] is a minimum provided [nu] squared [>=] 4AMgh. The method of small oscillations gave us the condition [nu] squared > 4AMgh, and indicated instability in the cases [nu] squared [=<] 4AMgh. The present criterion can also be applied to show that the steady precessional motions in which the axis has a constant inclination to the vertical are stable.
The question remains, as before, whether it is _essential_ for stability that V + [Kappa] should be a minimum. It appears that from the point of view of the theory of small oscillations it is not essential, and that there may even be stability when V + [Kappa] is a maximum. The precise conditions, which are of a somewhat elaborate character, have been formulated by Routh. An important distinction has, however, been established by Thomson and Tait, and by Poincare, between what we may call _ordinary_ or _temporary_ stability (which is stability in the above sense) and _permanent_ or _secular_ stability, which means stability when regard is had to possible dissipative forces called into play whenever the co-ordinates q1, q2, ... q_m vary. Since the total energy of the system at any instant is given (in the notation of Sec. 5) by an expression of the form [@] + V + [Kappa], where [@] cannot be negative, the argument of Thomson and Tait, given under MECHANICS, Sec. 23, for the statical question, shows that it is a necessary as well as a sufficient condition for secular stability that V + [Kappa] should be a minimum. When a system is "ordinarily" stable, but "secularly" unstable, the operation of the frictional forces is to induce a gradual increase in the amplitude of the free vibrations which are called into play by accidental disturbances.
There is a similar theory in relation to the constrained systems considered in Sec. 3 above. The equation (21) there given leads to the conclusion that for secular stability of any type of motion in which the velocities q`1, q`2, ... q`_n are zero it is necessary and sufficient that the function V - [Tau]0 should be a minimum.
The simplest possible example of this is the case of a particle at the lowest point of a smooth spherical bowl which rotates with constant angular velocity ([omega]) about the vertical diameter. This position obviously possesses "ordinary" stability. If a be the radius of the bowl, and [theta] denote angular distance from the lowest point, we have
V - [Tau]0 = mga(1 - cos [theta]) - 1/2m[omega] squareda squared sin squared [theta]; (2)
this is a minimum for [theta] = 0 only so long as [omega] squared < g/a. For greater values of [omega] the only position of "permanent" stability is that in which the particle rotates with the bowl at an angular distance cos^(-1) (g/[omega] squareda) from the lowest point. To examine the motion in the neighbourhood of the lowest point, when frictional forces are taken into account, we may take fixed ones, in a horizontal plane, through the lowest point. Assuming that the friction varies as the relative velocity, we have
x" = -p squaredx - k(x` + [omega]y), \ (3) y" = -p squaredy - k(y` - [omega]x), /
where p squared = g/a. These combine into
z" + kz` + (p squared - ik[omega])z = 0, (4)
where z = x + iy, i = [root]-1. Assuming z = Ce^([lambda]t), we find
[lambda] = -1/2k(1 [-+] [omega]/p) +- ip, (5)
if the square of k be neglected. The complete solution is then
x + iy = C1e^(-ss1t)e^(ipt) + C2e^(-ss2t)e^(-ipt), (6)
where ss1 = 1/2k(1 - [omega]/p), ss2 = 1/2k(1 + [omega]/p). (7)
This represents two superposed circular vibrations, in opposite directions, of period 2[pi]/p. If [omega] < p, the amplitude of each of these diminishes asymptotically to zero, and the position x = 0, y = 0 is permanently stable. But if [omega] > p the amplitude of that circular vibration which agrees in sense with the rotation [omega] will continually increase, and the particle will work its way in an ever-widening spiral path towards the eccentric position of secular stability. If the bowl be not spherical but ellipsoidal, the vertical diameter being a principal axis, it may easily be shown that the lowest position is permanently stable only so long as the period of the rotation is longer than that of the slower of the two normal modes in the absence of rotation (see MECHANICS, Sec. 13).
7. _Principle of Least Action._
Stationary Action.
The preceding theories give us statements applicable to the system at any one instant of its motion. We now come to a series of theorems relating to the whole motion of the system between any two configurations through which it passes, viz. we consider the actual motion and compare it with other imaginable motions, differing infinitely little from it, between the same two configurations. We use the symbol [delta] to denote the transition from the actual to any one of the hypothetical motions.
The best-known theorem of this class is that of _Least Action_, originated by P.L.M. de Maupertuis, but first put in a definite form by Lagrange. The "action" of a single particle in passing from one position to another is the space-integral of the momentum, or the time-integral of the _vis viva_. The action of a dynamical system is the sum of the actions of its constituent particles, and is accordingly given by the formula _ _ _ / / / A = [Sigma] | mvds = [Sigma] | mv squareddt = 2 | [Tau]dt. (1) _/ _/ _/
The theorem referred to asserts that the free motion of a conservative system between any two given configurations is characterized by the property
[delta]A = 0, (2)
provided the total energy have the same constant value in the varied motion as in the actual motion.
If t, t' be the times of passing through the initial and final configurations respectively, we have _ / t' [delta]A = [delta] | [Sigma]m(x` squared + y` squared + z` squared)dt _/t _ / t' = 2 | [delta][Tau]dt + 2[Tau]'[delta]t' + 2[Tau][delta]t, (3) _/t
since the upper and lower limits of the integral must both be regarded as variable. This may be written _ _ / t' / t' [delta]A = | [delta][Tau]dt + | [Sigma]m(x`[delta]x` + y`[delta]y` + z`[delta]z`)dt + 2[Tau]'[delta]t' - 2[Tau][delta]t _/t _/t _ _ _ / t' | | t' = | [delta][Tau]dt + | [Sigma]m (x`[delta]x + y`[delta]y + z`[delta]z | _/t |_ _| t _ / t' - | [Sigma]m(x"[delta]x + y"[delta]y + z"[delta]z)dt + 2[Tau]'[delta]t' - 2[Tau][delta]t. (4) _/t
Now, by d'Alembert's principle,
[Sigma]m( x"[delta]x + y"[delta]y + z"[delta]z ) = -[delta]V, (5)
and by hypothesis we have
[delta]([Tau] + V) = 0. (6)
The formula therefore reduces to _ _ | |t' [delta]A = | [Sigma]m (x`[delta]x + y`[delta]y + z`[delta]z) | + 2[Tau]'[delta]t' - 2[Tau][delta]t. (7) |_ _|t
Since the terminal configurations are unaltered, we must have at the lower limit
[delta]x + x`[delta]t = 0, [delta]y + y`[delta]t = 0, [delta]z + z`[delta]t = 0, (8)
with similar relations at the upper limit. These reduce (7) to the form (2).
The equation (2), it is to be noticed, merely expresses that the variation of A vanishes _to the first order_; the phrase _stationary action_ has therefore been suggested as indicating more accurately what has been proved. The action in the free path between two given configurations is in fact not invariably a minimum, and even when a minimum it need not be the _least possible_ subject to the given conditions. Simple illustrations are furnished by the case of a single particle. A particle moving on a smooth surface, and free from extraneous force, will have its velocity constant; hence the theorem in this case resolves itself into _ / [delta] | ds = 0, (9) _/
i.e. the path must be a geodesic line. Now a geodesic is not necessarily the _shortest_ path between two given points on it; for example, on the sphere a great-circle arc ceases to be the shortest path between its extremities when it exceeds 180 deg.. More generally, taking any surface, let a point P, starting from O, move along a geodesic; this geodesic will be a minimum path from O to P until P passes through a point O' (if such exist), which is the intersection with a consecutive geodesic through O. After this point the minimum property ceases. On an anticlastic surface two geodesics cannot intersect more than once, and each geodesic is therefore a minimum path between any two of its points. These illustrations are due to K.G.J. Jacobi, who has also formulated the general criterion, applicable to all dynamical systems, as follows:--Let O and P denote any two configurations on a natural path of the system. If this be the sole free path from O to P with the prescribed amount of energy, the action from O to P is a minimum. But if there be several distinct paths, let P vary from coincidence with O along the first-named path; the action will then cease to be a minimum when a configuration O' is reached such that two of the possible paths from O to O' coincide. For instance, if O and P be positions on the parabolic path of a projectile under gravity, there will be a second path (with the same energy and therefore the same velocity of projection from O), these two paths coinciding when P is at the other extremity (O', say) of the focal chord through O. The action from O to P will therefore be a minimum for all positions of P short of O'. Two configurations such as O and O' in the general statement are called conjugate _kinetic foci_. Cf. VARIATIONS, CALCULUS OF.
Before leaving this topic the connexion of the principle of stationary action with a well-known theorem of optics may be noticed. For the motion of a particle in a conservative field of force the principle takes the form _ / [delta] | vds = 0. (10) _/
On the corpuscular theory of light v is proportional to the refractive index mu of the medium, whence _ / [delta] | muds = 0. (11) _/
Hamiltonian principle.
In the formula (2) the energy in the hypothetical motion is prescribed, whilst the time of transit from the initial to the final configuration is variable. In another and generally more convenient theorem, due to Hamilton, the time of transit is prescribed to be the same as in the actual motion, whilst the energy may be different and need not (indeed) be constant. Under these conditions we have _ /t' [delta] | ([Tau] - V)dt = 0, (12) _/t
where t, t' are the prescribed times of passing through the given initial and final configurations. The proof of (12) is simple; we have _ _ _ /t' /t' /t' [delta] | ([Tau] - V)dt = | ([delta][Tau] - [delta]V)dt = | {[Sigma]m(x`[delta]x` + y`[delta]y` + z`[delta]z`) - [delta]V}dt _/t _/t _/t _ _ | |t' = | [Sigma]m(x`[delta]x + y`[delta]y + z`[delta]z) | |_ _|t _ /t' - | {[Sigma]m(x"[delta]x + y"[delta]y + z"[delta]z) + [delta]V}dt (13) _/t
The integrated terms vanish at both limits, since by hypothesis the configurations at these instants are fixed; and the terms under the integral sign vanish by d'Alembert's principle.
The fact that in (12) the variation does not affect the time of transit renders the formula easy of application in any system of co-ordinates. Thus, to deduce Lagrange's equations, we have
_ _ /t' /t'/dP[Tau] dP[Tau] dPV \ | ([delta][Tau]-[delta]V)dt = | ( -------[delta]q`1 + -------[delta]q1 + ... - ----[delta]q1 - ...)dt _/t _/t \ dPq`1 dPq1 dPq1 / _ _ | |t' = | p1[delta]q1 + p2[delta]q2 + ... | |_ _|t
_ _ _ /t'| / dP[Tau] dPV \ / dP[Tau] dPV\ | - | | (p`1 - ------- + ---- )[delta]q1 + (p`2 - ------- + -----)[delta]q2 + ...|dt. (14) _/t |_ \ dPq1 dPq1/ \ dPq2 dPq2/ _|
The integrated terms vanish at both limits; and in order that the remainder of the right-hand member may vanish it is necessary that the coefficients of [delta]q1, [delta]q2, ... under the integral sign should vanish for all values of t, since the variations in question are independent, and subject only to the condition of vanishing at the limits of integration. We are thus led to Lagrange's equation of motion for a conservative system. It appears that the formula (12) is a convenient as well as a compact embodiment of the whole of ordinary dynamics.
Extension to cyclic systems.
The modification of the Hamiltonian principle appropriate to the case of cyclic systems has been given by J. Larmor. If we write, as in Sec. 1 (25),
R = T - [kappa][chi]` - [kappa]'[chi]`' - [kappa]''[chi]`" - ..., (15)
we shall have _ /t' [delta] | (R - V)dt = 0, (16) _/t
provided that the variation does not affect the cyclic momenta [kappa], [kappa]', [kappa]", ..., and that the configurations at times t and t' are unaltered, so far as they depend on the palpable co-ordinates q1, q2, ... q_m. The initial and final values of the ignored co-ordinates will in general be affected.
To prove (16) we have, on the above understandings, _ _ /t' /t' [delta] | (R - V)dt = | ([delta][Tau] - [kappa][delta][chi]` - ... -[delta]V)dt _/t _/t _ /t' /dP[Tau] dP[Tau] \ = | ( -------[delta]q`1 + ... + -------[delta]q1 + ... - [delta]V )dt, (17) _/t \ dPq`1 dPq1 /
where terms have been cancelled in virtue of Sec. 5 (2). The last member of (17) represents a variation of the integral _ /t' | ([Tau] - V)dt _/t
on the supposition that [delta]X = 0, [delta]X' = 0, [delta]X" = 0, ... throughout, whilst [delta]q1, [delta]q2, [delta]q_m vanish at times t and t'; i.e. it is a variation in which the initial and final configurations are absolutely unaltered. It therefore vanishes as a consequence of the Hamiltonian principle in its original form.
Larmor has also given the corresponding form of the principle of least action. He shows that if we write _ / A = |(2[Tau] - [kappa][chi]` - [kappa]'[chi]`' - [kappa]"[chi]`" - ...)dt, (18) _/
then
[delta]A = 0, (19)
provided the varied motion takes place with the same constant value of the energy, and with the same constant cyclic momenta, between the same two configurations, these being regarded as defined by the palpable co-ordinates alone.
Sec. 8. _Hamilton's Principal and Characteristic Functions._
Principal function.
In the investigations next to be described a more extended meaning is given to the symbol [delta]. We will, in the first instance, denote by it an infinitesimal variation of the most general kind, affecting not merely the values of the co-ordinates at any instant, but also the initial and final configurations and the times of passing through them. If we put _ /t' S = | (T - V)dt, (1) _/t
we have, then, _ /t' [delta]S = (T' - V')[delta]t' - (T - V)[delta]t + | ([delta]T - [delta]V)dt _/t _ _ | |t' = (T' - V')[delta]t' - (T - V)[delta]t + |[Sigma]m(x`[delta]x + y`[delta]y + z`[delta]z)| (2) |_ _|t
Let us now denote by x' + [delta]x', y' + [delta]y', z' + [delta]z', the final co-ordinates (i.e. at time t' + [delta]t') of a particle m. In the terms in (2) which relate to the upper limit we must therefore write [delta]x' - x`'[delta]t', [delta]y' - y`'[delta]t', [delta]z' - z`'[delta]t' for [delta]x, [delta]y, [delta]z. With a similar modification at the lower limit, we obtain
[delta]S = - H[delta][tau] + [Sigma]m(x`'[delta]x' + y`'[delta]y' + z`'[delta]z') - [Sigma]m(x`[delta]x + y`[delta]y + z`[delta]z), (3)
where H(= T + V) is the constant value of the energy in the free motion of the system, and [tau](= t' - t) is the time of transit. In generalized co-ordinates this takes the form
[delta]S = - H[delta][tau] + p'1[delta]q'1 + p'2[delta]q'2 + ... - p1[delta]q1 - p2[delta]q2 - .... (4)
Now if we select any two arbitrary configurations as initial and final, it is evident that we can in general (by suitable initial velocities or impulses) start the system so that it will of itself pass from the first to the second in any prescribed time [tau]. On this view of the matter, S will be a function of the initial and final co-ordinates (q1, q2, ... and q'1, q'2, ...) and the time [tau], as independent variables. And we obtain at once from (4)
dPS dPS \ p'1 = -----, p'2 = -----, ..., | dPq'1 dPq'2 | > (5) dPS dPS | p1 = - ----, p2 = - ----, ..., | dPq1 dPq2 /
dPS and H = - -------. (6) dP[tau]
S is called by Hamilton the _principal function_; if its general form for any system can be found, the preceding equations suffice to determine the motion resulting from any given conditions. If we substitute the values of p1, p2, ... and H from (5) and (6) in the expression for the kinetic energy in the form [Tau]' (see Sec. 1), the equation
Tš + V = H (7)
becomes a partial differential equation to be satisfied by S. It has been shown by Jacobi that the dynamical problem resolves itself into obtaining a "complete" solution of this equation, involving n + 1 arbitrary constants. This aspect of the subject, as a problem in partial differential equations, has received great attention at the hands of mathematicians, but must be passed over here.
Characteristic function.
There is a similar theory for the function _ / A = 2 | Tdt = S + H[tau] (8) _/
It follows from (4) that
[delta]A = [tau][delta]H + p'1[delta]q'1 + p'2[delta]q'2 + ... - p1[delta]q1 - p2[delta]q2 - .... (9)
This formula (it may be remarked) contains the principle of "least action" as a particular case. Selecting, as before, any two arbitrary configurations, it is in general possible to start the system from one of these, with a prescribed value of the total energy H, so that it shall pass through the other. Hence, regarding A as a function of the initial and final co-ordinates and the energy, we find
dPA dPA \ p'1 = -----, p'2 = -----, ..., | dPq'1 dPq'2 | > (10) dPA dPA | p1 = - ----, p2 = - ----, ..., | dPq1 dPq2 /
dPA and [tau] = --- (11) dPH
A is called by Hamilton the _characteristic function_; it represents, of course, the "action" of the system in the free motion (with prescribed energy) between the two configurations. Like S, it satisfies a partial differential equation, obtained by substitution from (10) in (7).
The preceding theorems are easily adapted to the case of cyclic systems. We have only to write _ _ /t' /t' S = | (R - V)dt= | (T - [kappa][chi]` - [kappa]'[chi]`' - ... - V)dt (12) _/t _/t
in place of (1), and _ / A = | (2T - [kappa][chi]` - [kappa]'[chi]`' - ...)dt, (3) _/
in place of (8); cf. Sec. 7 ad fin. It is understood, of course, that in (12) S is regarded as a function of the initial and final values of the palpable co-ordinates q1, q2, ... q_m, and of the time of transit [tau], the cyclic momenta being invariable. Similarly in (13), A is regarded as a function of the initial and final values of q1, q2, ... q_m, and of the total energy H, with the cyclic momenta invariable. It will be found that the forms of (4) and (9) will be conserved, provided the variations [delta]q1, [delta]q2, ... be understood to refer to the palpable co-ordinates alone. It follows that the equations (5), (6) and (10), (11) will still hold under the new meanings of the symbols.
9. _Reciprocal Properties of Direct and Reversed Motions._
Lagrange's formula.
We may employ Hamilton's principal function to prove a very remarkable formula connecting any _two_ slightly disturbed natural motions of the system. If we use the symbols [delta] and [Delta] to denote the corresponding variations, the theorem is
d --[Sigma]([delta]p_r.[Delta]q_r - [Delta]p_r.[delta]q_r) = 0; (1) dt
or integrating from t to t',
[Sigma]([delta]p'_r.[Delta]q'_r - [Delta]q'_r.[delta]q'_r) = [Sigma]([delta]p_r.[Delta]q_r - [Delta]p_r.[delta]q_r). (2)
If for shortness we write
dP squaredS dP squaredS (r,s) = ----------, (r,s') = -----------, (3) dPq_rdPq_s dPq_rdPq'_s
we have
dPp_r = - [Sigma]_s(r,s)[delta]q_s - [Sigma]_s(r,s')[delta]q'_s (4)
with a similar expression for [Delta]p_r. Hence the right-hand side of (2) becomes
- [Sigma]_r{[Sigma]_s(r,s)[delta]q_s + [Sigma]_s(r,s')[delta]q'_s}[Delta]q_r + [Sigma]_r{[Sigma]_s(r,s)[Delta]q_s + [Sigma]_s(r,s')[Delta]q'_s}[delta]q_r = [Sigma]_r[Sigma]_s(r,s'){[delta]q_r.[Delta]q'_s - [Delta]q_r.[delta]q'_s}. (5)
The same value is obtained in like manner for the expression on the left hand of (2); hence the theorem, which, in the form (1), is due to Lagrange, and was employed by him as the basis of his method of treating the dynamical theory of _Variation of Arbitrary Constants_.
Helmholtz's reciprocal theorems.
The formula (2) leads at once to some remarkable reciprocal relations which were first expressed, in their complete form, by Helmholtz. Consider any natural motion of a conservative system between two configurations O and O' through which it passes at times t and t' respectively, and let t' - t = [tau]. As the system is passing through O let a small impulse [delta]p_r be given to it, and let the consequent alteration in the co-ordinate q_s after the time [tau] be [delta]q'_s. Next consider the _reversed_ motion of the system, in which it would, if undisturbed, pass from O' to O in the same time [tau]. Let a small impulse [delta]p'_s be applied as the system is passing through O', and let the consequent change in the co-ordinate q_r after a time [tau] be [delta]q_r. Helmholtz's first theorem is to the effect that
[delta]q_r : [delta]p'_s = [delta]q'_s : [delta]p_r. (6)
To prove this, suppose, in (2), that all the [delta]q vanish, and likewise all the [delta]p with the exception of [delta]p_r. Further, suppose all the [Delta]q' to vanish, and likewise all the [Delta]p' except [Delta]p'_s, the formula then gives
[delta]p_r.[Delta]q_r = - [Delta]p'_s.[delta]q'_s, (7)
which is equivalent to Helmholtz's result, since we may suppose the symbol [Delta] to refer to the reversed motion, provided we change the signs of the [Delta]p. In the most general motion of a top (MECHANICS, Sec. 22), suppose that a small impulsive couple about the vertical produces after a time [tau] a change [delta][theta] in the inclination of the axis, the theorem asserts that in the reversed motion an equal impulsive couple in the plane of [theta] will produce after a time [tau] a change [delta][psi], in the azimuth of the axis, which is equal to [delta][theta]. It is understood, of course, that the couples have no components (in the generalized sense) except of the types indicated; for instance, they may consist in each case of a force applied to the top at a point of the axis, and of the accompanying reaction at the pivot. Again, in the corpuscular theory of light let O, O' be any two points on the axis of a symmetrical optical combination, and let V, V' be the corresponding velocities of light. At O let a small impulse be applied perpendicular to the axis so as to produce an angular deflection [delta][theta], and let ss' be the corresponding lateral deviation at O'. In like manner in the reversed motion, let a small deflection [delta][theta]' at O' produce a lateral deviation ss at O. The theorem (6) asserts that
ss ss' ----------------- = ---------------, (8) V'[delta][theta]' V[delta][theta]
or, in optical language, the "apparent distance" of O from O' is to that of O' from O in the ratio of the refractive indices at O' and O respectively.
Helmholtz's second reciprocal theorem.
In the second reciprocal theorem of Helmholtz the configuration O is slightly varied by a change [delta]q_r in one of the co-ordinates, the momenta being all unaltered, and [delta]q'_s is the consequent variation in one of the momenta after time [tau]. Similarly in the reversed motion a change [delta]p'_s produces after time [tau] a change of momentum [delta]p_r. The theorem asserts that
[delta]p'_s : [delta]q_r = [delta]p_r : [delta]q'_s (9)
This follows at once from (2) if we imagine all the [delta]p to vanish, and likewise all the [delta]q save [delta]q_r, and if (further) we imagine all the [Delta]p' to vanish, and all the [Delta]q' save [Delta]q'_s. Reverting to the optical illustration, if F, F', be principal foci, we can infer that the convergence at F' of a parallel beam from F is to the convergence at F of a parallel beam from F' in the inverse ratio of the refractive indices at F' and F. This is equivalent to Gauss's relation between the two principal focal lengths of an optical instrument. It may be obtained otherwise as a particular case of (8).
We have by no means exhausted the inferences to be drawn from Lagrange's formula. It may be noted that (6) includes as particular cases various important reciprocal relations in optics and acoustics formulated by R.J.E. Clausius, Helmholtz, Thomson (Lord Kelvin) and Tait, and Lord Rayleigh. In applying the theorem care must be taken that in the reversed motion the reversal is complete, and extends to every velocity in the system; in particular, in a cyclic system the cyclic motions must be imagined to be reversed with the rest. Conspicuous instances of the failure of the theorem through incomplete reversal are afforded by the propagation of sound in a wind and the propagation of light in a magnetic medium.
It may be worth while to point out, however, that there is no such limitation to the use of Lagrange's formula (1). In applying it to cyclic systems, it is convenient to introduce conditions already laid down, viz. that the co-ordinates q_r are the palpable co-ordinates and that the cyclic momenta are invariable. Special inference can then be drawn as before, but the interpretation cannot be expressed so neatly owing to the non-reversibility of the motion.
AUTHORITIES.--The most important and most accessible early authorities are J.L. Lagrange, _Mecanique analytique_ (1st ed. Paris, 1788, 2nd ed. Paris, 1811; reprinted in _Oeuvres_, vols. xi., xii., Paris, 1888-89); Hamilton, "On a General Method in Dynamics," _Phil. Trans._ 1834 and 1835; C.G.J. Jacobi, _Vorlesungen ueber Dynamik_ (Berlin, 1866, reprinted in _Werke_, Supp.-Bd., Berlin, 1884). An account of the extensive literature on the differential equations of dynamics and on the theory of variation of parameters is given by A. Cayley, "Report on Theoretical Dynamics," _Brit. Assn. Rep._ (1857), _Mathematical Papers_, vol. iii. (Cambridge, 1890). For the modern developments reference may be made to Thomson and Tait, _Natural Philosophy_ (1st ed. Oxford, 1867, 2nd ed. Cambridge, 1879); Lord Rayleigh, _Theory of Sound_, vol. i. (1st ed. London, 1877; 2nd ed. London, 1894); E.J. Routh, _Stability of Motion_ (London, 1877), and _Rigid Dynamics_ (4th ed. London, 1884); H. Helmholtz, "Ueber die physikalische Bedeutung des Prinzips der kleinsten Action," _Crelle_, vol. c., 1886, reprinted (with other cognate papers) in _Wiss. Abh._ vol. iii. (Leipzig, 1895); J. Larmor, "On Least Action," _Proc. Lond. Math. Soc._ vol. xv. (1884); E.T. Whittaker, _Analytical Dynamics_ (Cambridge, 1904). As to the question of stability, reference may be made to H. Poincare, "Sur l'equilibre d'une masse fluide animee d'un mouvement de rotation" _Acta math._ vol. vii. (1885); F. Klein and A. Sommerfeld, _Theorie des Kreisels_, pts. 1, 2 (Leipzig, 1897-1898); A. Lioupanoff and J. Hadamard, _Liouville_, 5me serie, vol. iii. (1897); T.J.I. Bromwich, Proc. Lond. Math. Soc. vol. xxxiii. (1901). A remarkable interpretation of various dynamical principles is given by H. Hertz in his posthumous work _Die Prinzipien der Mechanik_ (Leipzig, 1894), of which an English translation appeared in 1900. (H. Lb.)
DYNAMITE (Gr. [Greek: dynamis], power), the name given to several explosive preparations containing nitroglycerin (q.v.) which are almost exclusively used for blasting purposes. The first practical application of nitroglycerin in this way was made by A. Nobel in 1863. He soaked gunpowder with the liquid and fired the gunpowder by an ordinary fuse. Later he found that nitroglycerin could be detonated by the explosion of several materials such as fulminate of mercury, the use of which as a detonator he patented in 1867. In 1866-1867 he experimented with charcoal and other substances, and found the infusorial earth known as kieselguhr, which consists mainly of silica (nearly 95%), eminently adapted to the purpose, as it was inert, non-combustible, and after a little heating and preparation very porous, retaining a large amount of nitroglycerin as water is held in a sponge, without very serious exudation on standing. This kieselguhr dynamite is generally made by incorporating three parts of nitroglycerin with one part of the dry earth, the paste being then formed into cylindrical cartridges. This work is done by hand. Generally a small percentage of the kieselguhr is replaced by a mixture containing sodium and ammonium carbonates, talc and ochre. This product is known as dynamite No. 1. Disabilities attaching to kieselguhr dynamite are that when placed in water the nitroglycerin is liable to be exuded or displaced, also that, like nitroglycerin itself, it freezes fairly easily and thawing the frozen cartridges is a dangerous operation. Other substances, e.g. kaolin, tripoli, magnesia alba (magnesium carbonate), alumina, sugar, charcoal, some powdered salts and mixtures of sawdust and salts, have been shown to be absorbents more or less adapted to the purpose of making a dynamite. Charcoal from cork is said to absorb about 90% of its weight of nitroglycerin. With the idea of obtaining greater safety, mixtures have been made of nitroglycerin with wood fibre, charcoal and metallic nitrates. Lithofracteur, for instance, consists of 50% nitroglycerin and a mixture of prepared sawdust, kieselguhr and barium nitrate. Carbonite contains 25% of nitroglycerin, the remainder being a mixture of wood-meal and alkali nitrates, with about 1% of sulphur. Dualin, atlas dynamite and potentite are other modifications.
A convenient form in which nitroglycerin can be made up for blasting purposes, especially in wet ground, is the gelatinous material obtained by the action of nitroglycerin, either alone or with the help of solvents, on low-grade or soluble gun-cottons. It is known as blasting gelatin, and was first made by Nobel by incorporating 6 or 7% of low nitrated cellulose (collodion cotton or soluble gun-cotton) with slightly warmed nitroglycerin. The result is a transparent plastic material, of specific gravity 1.5 to 1.6, which may be kept under water for a long time without appreciable change. It is less sensitive to detonation than ordinary dynamite, and although its explosion is slightly slower it is more powerful than dynamite and much superior to the liquid nitroglycerin. Blasting gelatin also freezes and is sensitive to percussion in this state. Camphor and other substances have been added to blasting gelatin to render it more solid and less sensitive. Some modifications of blasting gelatin, e.g. gelignite, contain wood-meal and such oxygen-containing salts as potassium nitrate. Experience has conclusively shown that dynamites are more satisfactory, quicker, and more intense in action than liquid nitroglycerin.
To prevent nitroglycerin and some of the forms of dynamite from freezing it has been proposed to add to them small quantities of either monochlor-dinitroglycerin or of a nitrated poly-glycerin. The former is obtained by first acting upon glycerin with hydrogen chloride to produce _u-_chlorhydrin or chlor-propylene glycol, C3H7O2Cl, which is then nitrated as in the case of glycerin. The latter is obtained by heating glycerin for six or seven hours to about 300 deg. C., whereby water is split off in such manner that a diglycerin C6H14O5, for the most part, results. This on nitration in the usual manner gives a product C6H{10}N4O{13}, which burns and explodes in a similar manner to ordinary nitroglycerin, but is less sensitive and does not so easily freeze. The mono- and di-nitrates of glycerin have also been proposed as additions to ordinary nitroglycerin (q.v.) for the same purpose. (W. R. E. H.)
DYNAMO (a shortened form of "dynamo-electric machine," from Gr. [Greek: dynamis], power), a machine for converting mechanical into electrical energy.
The dynamo ranks with the telegraph and telephone as one of the three striking applications of electrical and magnetic science to which the material progress that marked the second half of the 19th century was in no small measure due. Since the discovery of the principle of the dynamo by Faraday in 1831 the simple model which he first constructed has been gradually developed into the machines of 5000 horse-power or more which are now built to meet the needs of large cities for electric lighting and power, while at the same time the numbers of dynamos in use have increased almost beyond estimate. Yet such was the insight of Faraday into the fundamental nature of the dynamo that the theory of its action which he laid down has remained essentially unchanged. His experiments on the current which was set up in a coil of wire during its movement across the poles of a magnet led naturally to the explanation of induced electromotive force as caused by the linking or unlinking of magnetic lines of flux with an electric circuit. For the more definite case of the dynamo, however, we may, with Faraday, make the transition from line-linkage to the equivalent conception of "line-cutting" as the source of E.M.F.--in other words, to the idea of electric conductors "cutting" or intersecting[1] the lines of flux in virtue of relative motion of the magnetic field and electric circuit. On the 28th of October 1831 Faraday mounted a copper disk so that it could be rotated edgewise between the poles of a permanent horse-shoe magnet. When so rotated, it cut the lines of flux which passed transversely through its lower half, and by means of two rubbing contacts, one on its periphery and the other on its spindle, the circuit was closed through a galvanometer, which indicated the passage of a continuous current so long as the disk was rotated (fig. 1). Thus by the invention of the first dynamo Faraday proved his idea that the E.M.F. induced through the interaction of a magnetic field and an electric circuit was due to the passage of a portion of the electric circuit _across_ the lines of flux, or vice versa, and so could be maintained if the cutting of the lines were made continuous.[2] In comparison with Faraday's results, the subsequent advance is to be regarded as a progressive perfecting of the mechanical and electro-magnetic design, partly from the theoretical and partly from the practical side, rather than as modifying or adding to the idea which was originally present in his mind, and of which he already saw the possibilities.
A dynamo, then, is a machine in which, by means of continuous relative motion, an electrical conductor or system of conductors forming part of a circuit is caused to cut the lines of a magnetic field or fields; the cutting of the magnetic flux induces an electromotive force in the conductors, and when the circuit is closed a current flows, whereby mechanical energy is converted into electrical energy.
Little practical use could be made of electrical energy so long as its only known sources were frictional machines and voltaic batteries. The cost of the materials for producing electrical currents on a large scale by chemical action was prohibitive, while the frictional machine only yielded very small currents at extremely high potentials. In the dynamo, on the other hand, electrical energy in a convenient form could be cheaply and easily obtained by mechanical means, and with its invention the application of electricity to a wide range of commercial purposes became economically possible. As a converter of energy from one form to another it is only surpassed in efficiency by another electrical appliance, namely, the transformer (see TRANSFORMERS). In this there is merely conversion of electrical energy at a high potential into electrical energy at a low potential, or vice versa, but in the dynamo the mechanical energy which must be applied to maintain the relative movement of magnetic field and conductor is absorbed, and reappears in an electrical form. A true transformation takes place, and the proportion which the rate of delivery of electrical energy bears to the power absorbed, or in other words the _efficiency_, is the more remarkable. The useful return or "output" at the terminals of a large machine may amount to as much as 95% of the mechanical energy which forms the "input." Since it needs some prime mover to drive it, the dynamo has not made any direct addition to our sources of energy, and does not therefore rank with the primary battery or oil-engine, or even the steam-engine, all of which draw their energy more immediately from nature. Yet by the aid of the dynamo the power to be derived from waterfalls can be economically and conveniently converted into an electrical form and brought to the neighbouring factory or distant town, to be there reconverted by motors into mechanical power. Over any but very short distances energy is most easily transmitted when it is in an electrical form, and turbine-driven dynamos are very largely and successfully employed for such transmission. Thus by conducing to the utilization of water-power which may previously have had but little value owing to its disadvantageous situation, the dynamo may almost be said to have added another to our available natural resources.
The two essential parts of the dynamo, as required by its definition, may be illustrated by the original disk machine of Faraday. They are (1) the _iron magnet_, between the poles of which a magnetic field exists, and (2) the _electrical conductors_, represented by the rotating copper disk. The sector of the disk cutting the lines of the field forms part of a closed electric circuit, and has an E.M.F. induced in it, by reason of which it is no longer simply a conductor, but has become "active." In its more highly developed form the simple copper disk is elaborated into a system of many active wires or bars which form the "winding," and which are so interconnected as to add up their several E.M.F.'s. Since these active wires are usually mounted on an iron structure, which may be likened to the keeper or "armature" of a magnet rotating between its poles, the term "armature" has been extended to cover not only the iron core, but also the wires on it, and when there is no iron core it is even applied to the copper conductors themselves. In the dynamo of Faraday the "armature" was the rotating portion, and such is the case with modern continuous-current dynamos; in alternators, however, the magnet, or a portion of it, is more commonly rotated while the armature is stationary. It is in fact immaterial to the action whether the one or the other is moved, or both, so long as their relative motion causes the armature conductors to cut the magnetic flux. As to the ultimate reason why an E.M.F. should be thereby induced, physical science cannot as yet yield any surer knowledge than in the days of Faraday.[3] For the engineer, it suffices to know that the E.M.F. of the dynamo is due to the cutting of the magnetic flux by the active wires, and, further, is proportional to the rate at which the lines are cut.[4]
The equation of the _electromotive force_ which is required in order to render this statement quantitative must contain three factors, namely, the density of the flux in the air-gap through which the armature conductors move, the active length of these wires, and the speed of their movement. For given values of the first and third factors and a single straight wire moved parallel to itself through a uniform field, the maximum rate of cutting is evidently obtained when the three directions of the lines of the conductor's length and of the relative motion are respectively at right angles to each other, as shown by the three co-ordinate axes of fig. 2. The E.M.F. of the single wire is then
E = B_gLV x 10^(-8) volts (1)
where B_g is the density of the flux within the air-gap expressed in C.G.S. lines per square centimetre, L is the active length of the conductor within the field in centimetres, and V is the velocity of movement in centimetres per second. Further, the direction in which the E.M.F. has the above maximum value is along the length of the conductor, its "sense" being determined by the direction of the movement[5] in relation to the direction of the field.
The second fundamental equation of the dynamo brings to light its mechanical side, and rests on H.C. Oersted's discovery of the interaction of a magnetic field and an electric current. If a straight electric conductor through which a current is passing be so placed in a magnetic field that its length is not parallel to the direction of the lines of flux, it is acted on by a force which will move it, if free, in a definite direction relatively to the magnet; or if the conductor is fixed and the magnet is free, the latter will itself move in the opposite direction. Now in the dynamo the active wires are placed so that their length is at right angles to the field; hence when they are rotated and an electric current begins to flow under the E.M.F. which they induce, a mutual force at once arises between the copper conductors and the magnet, and the direction of this force must by Lenz's law be opposed to the direction of the movement. Thus as soon as the disk of fig. 1 is rotated and its circuit is closed, it experiences a mechanical pull or drag which must be overcome by the force applied to turn the disk. While the magnet must be firmly held so as to remain stationary, the armature must be of such mechanical construction that its wires can be forcibly driven through the magnetic field against the mutual pull. This law of electrodynamic action may be quantitatively stated in an _equation of mechanical force_, analogous to the equation (I.) of electromotive force, which states the law of electromagnetic induction. If a conductor of length L cm., carrying a current C amperes, is immersed in a field of uniform density B_g, and the length of the conductor is at right angles to the direction of the lines, it is acted on by a force
F = B_gLC x 10^(-1) dynes, (2)
and the direction of this force is at right angles to the conductor and to the field. The rate at which electrical energy is developed, when this force is overcome by moving the conductor as a dynamo through the field, is EC = B_gLVC x 10^(-8) watts, whence the equality of the mechanical power absorbed and the electrical power developed (as required by the law of the conservation of energy) is easily established. The whole of this power is not, however, available at the terminals of the machine; if R_a be the resistance of the armature in ohms, the passage of the current C_a through the armature conductors causes a drop of pressure of C_aR_a volts, and a corresponding loss of energy in the armature at the rate of C_a squaredR_a watts. As the resistance of the external circuit R_e is lowered, the current C = E_a/(R_e + R_a) is increased. The increase of the current is, however, accompanied by a progressive increase in the loss of energy over the armature, and as this is expended in heating the armature conductors, their temperature may rise so much as to destroy the insulating materials with which they are covered. Hence the temperature which the machine may be permitted to attain in its working is of great importance in determining its output, the current which forms one factor therein being primarily limited by the heating which it produces in the armature winding. The lower the resistance of the armature, the less the rise of its temperature for a given current flowing through it; and the reason for the almost universal adoption of copper as the material for the armature conductors is now seen to lie in its high conductivity.[6]
Since the voltage of the dynamo is the second factor to which its output is proportional, the conditions which render the induced E.M.F. a maximum must evidently be reproduced as far as possible in practice, if the best use is to be made of a given mass of iron and copper. The first problem, therefore, in the construction of the dynamo is the disposition of the wires and field in such a manner that the three directions of field, length of active conductors, and movement are at right angles to one another, and so that the relative motion is continuous. Reciprocating motion, such as would be obtained by direct attachment of the conductors to the piston of a steam-engine, has been successfully employed only in the special case of an "oscillator,"[7] producing a small current very rapidly changing in direction. Rotary motion is therefore universally adopted, and with this two distinct cases arise. Either (A) the active length of the wire is parallel to the axis of rotation, or (B) it is at right angles to it.
(A) If a conductor is rotated in the gap between the poles of a horse-shoe magnet, and these poles have plane parallel faces opposing one another as in fig. 3, not only is the density of the flux in the interpolar gap small, but the direction of movement is not always at right angles to the direction of the lines, which for the most part pass straight across from one opposing face to the other. When the conductor is midway between the poles (i.e. either at its highest or lowest point), it is at this instant sliding along the lines and does not cut them, so that its E.M.F. is zero. Taking this position as the starting-point, as the conductor moves round, its rate of line-cutting increases to a maximum when it has moved through a right angle and is opposite to the centre of a pole-face (as in fig. 3), from which point onward the rate decreases to zero when it has moved through 180 deg.. Each time the conductor crosses a line drawn symmetrically through the gap between the poles and at right angles to the axis of rotation, the E.M.F. along its length is reversed in direction, since the motion relatively to the direction of the field is reversed. If the ends of the active conductor are electrically connected to two collecting rings fixed upon, but insulated from, the shaft, two stationary brushes bb can be pressed on the rings so as to make a sliding contact. An external circuit can then be connected to the brushes, which will form the "terminals" of the machine, the periodically reversed or alternating E.M.F. induced in the active conductor will cause an alternating current to flow through conductor and external circuit, and the simplest form of "alternator" is obtained. If the field cut by the straight conductor is of uniform density, and all the lines pass straight across from one pole-face to the other (both of which assumptions are approximately correct), a curve connecting the instantaneous values of the E.M.F. as ordinates with time or degrees of angular movement as abscissae (as shown at the foot of fig. 3), will, if the speed of rotation be uniform, be a sine curve. If, however, the conductor is mounted on an iron cylinder (fig. 4),[8] a sufficient margin being allowed for mechanical clearance between it and the poles, not only will the reluctance of the magnetic circuit be reduced and the total flux and its density in the air-gap B_g be thereby increased, but the path of the lines will become nearly radial, except at the "fringe" near the edges of the pole-tips; hence the relative directions of the movement and of the lines will be continuously at right angles. The shape of the E.M.F. curve will then be as shown in fig. 4--flat-topped, with rounded corners rapidly sloping down to the zero line.
But a single wire cannot thus be made to give more than a few volts, and while dynamos for voltages from 5 to 10 are required for certain purposes, the voltages in common use range from 100 to 10,000. It is therefore necessary to connect a number of such wires in series, so as to form an "armature winding." If several similar conductors are arranged along the length of the iron core parallel to the first (fig. 5), the E.M.F.'s generated in the conductors which at any moment are under the same pole are similarly directed, and are opposite to the directions of the E.M.F.'s in the conductors under the other pole (cf fig. 5 where the dotted and crossed ends of the wires indicate E.M.F.'s directed respectively towards and away from the observer). Two distinct methods of winding thence arise, the similarity of the E.M.F.'s under the same pole being taken advantage of in the first, and the opposite E.M.F.'s under N and S poles in the second.
1. The first, or _ring_-winding, was invented by Dr Antonio Pacinotti of Florence[9] in 1860, and was subsequently and independently reintroduced in 1870[10] by the Belgian electrician, Zenobe Theophile Gramme, whence it is also frequently called the "Gramme" winding. By this method the farther end of conductor 1 (fig. 5) is joined in series to the near end of conductor 2; this latter lies next to it on the surface of the core or immediately above it, so that both are simultaneously under the same pole-piece. For this series connexion to be possible, the armature core must be a hollow cylinder, supported from the shaft on an open non-magnetic spider or hub, between the arms of which there is room for the internal wire completing the loop (fig. 6). The end of one complete loop or turn embracing one side of the armature core thus forms the starting-point for another loop, and the process can be continued if required to form a coil of two or more turns. In the ring armature the iron core serves the double purpose of conducting the lines across from one pole to the other, and also of shielding from the magnetic flux the hollow interior through which the connecting wires pass. Any lines which leak across the central space are cut by the internal wires, and the direction of cutting is such that the E.M.F. caused thereby opposes the E.M.F. due to the active conductors proper on the external surface. If, however, the section of iron in the core be correctly proportioned, the number of lines which cross the interior will bear but a small ratio to those which pass entirely through the iron, and the counter E.M.F. of the internal wires will become very small; they may then be regarded simply as connectors for joining the external active wires in series.
2. The second or _drum_ method was used in the original "shuttle-wound" armatures invented by Dr Werner von Siemens in 1856, and is sometimes called the "Siemens" winding. The farther end of conductor 1 (fig. 5) is joined by a connecting wire to the farther end of another conductor 2' situated nearly diametrically opposite on the other side of the core and under the opposite pole-piece. The near end of the complete loop or turn is then brought across the end of the core, and can be used as the starting-point for another loop beginning with conductor 2, which is situated by the side of the first conductor. The iron core may now be solid from the surface to the shaft, since no connecting wires are brought through the centre, and each loop embraces the entire armature core (fig. 7). By the formation of two loops in the ring armature and of the single loop in the drum armature, two active wires are placed in series; the curves of instantaneous E.M.F. are therefore similar in shape to that of the single wire (fig. 4), but with their ordinates raised throughout to double their former height, as shown at the foot of fig. 6.
Next, if the free ends of either the ring or drum loops, instead of being connected to two collecting rings, are attached to the two halves of a split-ring insulated from the shaft (as shown in fig. 7 in connexion with a drum armature), and the stationary brushes are so set relatively to the loops that they pass over from the one half of the split-ring to the other half at the moment when the loops are passing the centre of the interpolar gap, and so are giving little or no E.M.F., each brush will always remain either positive or negative. The current in the external circuit attached to the brushes will then have a constant direction, although the E.M.F. in the active wires still remains alternating; the curve of E.M.F. obtained at the brushes is thus (as in fig. 7) entirely above the zero line. The first dynamo of H. Pixii,[11] which immediately followed Faraday's discovery, gave an alternating current, but in 1832[12] the alternator was converted into a machine giving a _unidirected current_ by the substitution of a rudimentary "commutator" in place of mercury collecting cups.
(B) So far the length of the active wires has been parallel to the axis of rotation, but they may equally well be arranged perpendicularly thereto. The poles will then have plane faces and the active wires will be disposed with their length approximately radial to the axis of the shaft. In order to add their E.M.F.'s in series, two types of winding may be employed, which are precisely analogous in principle to the ring and drum windings under arrangement (A).
3. The _discoidal_ or flat-ring armature is equivalent to a ring of which the radial depth greatly exceeds the length, with the poles presented to one side of the ring instead of embracing its cylindrical surface. A similar set of poles is also presented to the opposite side of the ring, like poles being opposite to one another, so that in effect each polar surface is divided into two halves, and the groups of lines from each side bifurcate and pass circumferentially through the armature core to issue into the adjacent poles of opposite sign.
4. In the _disk_ machine, no iron core is necessary for the armature, the two opposite poles of unlike sign being brought close together, leaving but a short path for the lines in the air-gap through which the active wires are rotated.
If the above elementary dynamos are compared with fig. 1, it will be found that they all possess a distinctive feature which is not present in the original disk machine of Faraday. In the four types of machine above described each active wire in each revolution first cuts the group of lines forming a field in one direction, and then cuts the same lines again in the opposite direction relatively to the sense of the lines, so that along the length of the wire the E.M.F. alternates in direction. But in the dynamo of fig. 1 the sector of the copper disk which is at any moment moving through the magnetic field and which forms the single active element is always cutting the lines in the same manner, so that the E.M.F. generated along its radial length is continuous and unchanged in direction. This radical distinction differentiates the two classes of _heteropolar_ and _homopolar_ dynamos, Faraday's disk machine of fig. 1 being the type of the latter class. In it the active element may be arranged either parallel or at right angles to the axis of rotation; but in both cases, in order to increase the E.M.F. by placing two or more elements in series, it becomes necessary either (1) to employ some form of sliding contact by which the current may be collected from the end of one active element and passed round a connecting wire into the next element without again cutting the field in the reverse direction, or (2) to form on the armature a loop of which each side is alternately active and inactive. The first method limits the possibilities of the homopolar machine so greatly when large currents and high voltages are required that it is now only used in rare instances, as e.g. occasionally in dynamos driven by steam-turbines which have a very high speed of rotation. The second alternative may be carried into effect with any of the four methods of armature winding, but is practically confined to the drum and disk types. In its drum form the field is divided into two or more projecting poles, all of the same sign, with intervening neutral spaces of equal width, and the span of the loop in the direction of rotation is at least equal to the width of a polar projection, as in fig. 8, where two polar projections are shown. Each side of the loop then plays a dual part; it first cuts the lines of one polar projection and generates an E.M.F., and next becomes an inactive connecting wire, while the action is taken up by the opposite side of the loop which has previously served as a connector but now cuts the lines of the next polar projection. The E.M.F. is thus always in the same direction along the side which is at any moment active, but alternates round the loop as a whole, and the distinctive peculiarity of the homopolar machine, so soon as any form of "winding" is introduced into its armature, is lost. It results that the homopolar principle, which would prima facie appear specially suitable for the generation of a unidirectional E.M.F. and continuous current, can seldom be used for this purpose and is practically confined to alternators. It may therefore be said that in almost all dynamos, whether they supply an alternating or a continuous current in the external circuit, the E.M.F. and current in the armature are alternating.
Ring winding was largely employed in early continuous-current dynamos and also in the alternators of Gramme and H. Wilde, and later of Auguste de Meritens. Disk winding was also successfully introduced for alternators, as in the magneto-machines of Nollet (1849) and the alternators of Wilde (1866) and Siemens (1878), and its use was continued in the machines of W.M. Mordey and S.Z. Ferranti. But although the ring, discoidal-ring and disk methods of winding deserve mention from their historical importance, experience has shown that drum winding possesses a marked superiority for both electrical and manufacturing reasons; the three former methods have in fact been practically discarded in its favour, so that the drum method will hereafter alone be considered.
The drum coil, composed of several loops wound side by side, may therefore be regarded as the constituent active element out of which the armature winding of the modern dynamo is developed. Its application to the multipolar machine is easily followed from fig. 9, which illustrates the heteropolar type of dynamo. The span of the loops, which is nearly 180 deg. or across the diameter of the two-pole machine, is reduced approximately to 90 deg. in the four-pole or to 60 deg. in the six-pole machine and so on, the curvature of the coil becoming gradually less as the number of poles is increased. The passage of a coil through two magnetic fields of opposite direction yields a complete wave of E.M.F., such as is shown in fig. 6, and the time in seconds taken to pass through such a complete cycle is the "period" of the alternating E.M.F. The number of complete periods through which the E.M.F. of the coil passes per second is called the "periodicity" or "frequency" of the machine. In the bipolar machine this is equal to the number of revolutions per second, and in the multipolar machine it is equal to the number of pairs of fields through which the coil passes in one second; hence in general the periodicity is pN/60, where N = the number of revolutions per minute and p = the number of pairs of poles, and this holds true of the E.M.F. and current round the coil, even though the E.M.F. and current furnished to the external circuit may be rendered unidirectional or continuous. The only difference on this point is that in the continuous-current machine the poles are usually fewer than in the alternator, and the periodicity is correspondingly lower. Thus in the former case the number of poles ranges from 2 to 12 and the usual frequencies from 5 to 20; but with alternators the frequencies in commercial use range from 25 to 120, and in large machines driven by slow-speed engines the number of poles may even be as high as 96.
The drum coil may be applied either to the external surface of a rotating armature, the field-magnet being external and stationary (fig. 9), or to the internal surface of a stationary armature (fig. 10), the field-magnet being internal and rotating. While the former combination is universally adopted in the continuous-current dynamo, the latter is more usual in the modern alternator. In either case the iron armature core must be "laminated"; the passage of the lines of the field across its surface sets up E.M.F.'s which are in opposite directions under poles of opposite sign, so that if the core were a solid mass a current-sheet would flow along its surface opposite to a pole, and complete its circuit by passing through the deeper layers of metal or by returning in a sheet under a pole of opposite sign. Such "eddy-currents" can be practically avoided by dividing the metal core into laminations at right angles to the length of the active wires which are themselves arranged to secure the greatest rate of line-cutting and maximum E.M.F. The production of the eddy-current E.M.F. is not thereby prevented, but the paths of the eddy-currents are so broken up that the comparatively high resistance with which they meet reduces their amount very greatly. The laminae must be lightly insulated from one another, right up to their edges, so that the E.M.F.'s which still act across their thickness will not be added up along the length of the core, but will only produce extremely small currents circulating through the interior of the separate laminations. Each thin iron plate is either coated with an insulating varnish or has one of its sides covered with a sheet of very thin paper; the thickness of the laminae is usually about one-fortieth of an inch, and if this is not exceeded the rate at which energy is dissipated by eddy-currents in the core is so far reduced that it does not seriously impair the efficiency of the machine.
Lastly, the drum coils may be either attached to the surface of a smooth armature core (fig. 9, I.), or may be wound through holes formed close to the periphery of the core, or may be embedded in the slots between projecting iron teeth (figs. 9 [II.] and 10). Originally employed by Antonio Pacinotti in connexion with ring winding, the toothed armature was after some considerable use largely discarded in favour of the smooth core; it has, however, been reintroduced with a fuller understanding of the special precautions necessitated in its design, and it is now so commonly used that it may be said to have superseded the smooth-surface armature.
Not only does the toothed armature reduce the length of the air-gap to the minimum permitted by mechanical and magnetic considerations, and furnish better mechanical protection to the armature coils, but it also ensures the positive holding of the active wires against the mechanical drag which they experience as they pass through the magnetic field. Further, the active wires in the toothed armature are relieved of a large proportion of this mechanical drag, which is transferred to the iron teeth. The lines of the field, after passing through the air-gap proper, divide between the teeth and the slots in proportion to their relative permeances. Hence at any moment the active wires are situated in a weak field, and for a given armature current the force on them is only proportional to this weak field. This important result is connected with the fact that when the armature is giving current the distribution of the lines over the face of each tooth is distorted, so that they become denser on the "trailing" side than on the "leading" side;[13] the effect of the non-uniform distribution acting on all the teeth is to produce a magnetic drag on the armature core proportional to the current passing through the wires, so that the total resisting force remains the same as if the armature had a smooth core. The amount by which the stress on the active wires is reduced entirely depends upon the degree to which the teeth are saturated, but, since the relative permeability of iron even at a flux density of 20,000 lines per sq. cm. is to that of air approximately as 33:1, the embedded wires are very largely relieved of the driving stress. An additional gain is that solid bars of much greater width can be used in the toothed armature than on a smooth core without appreciable loss from eddy-currents within their mass.
A disadvantage of the slotted core is, however, that it usually necessitates the lamination of the pole-pieces. If the top of the slot is open, and its width of opening is considerably greater than the length of the air-gap from the iron of the pole-face to the surface of the teeth, the lines become unequally distributed not only at the surface of the teeth, but also at the face of the pole-pieces; and this massing of the lines into bands causes the density at the pole-face to be rhythmically varied as the teeth pass under it. No such variation can take place in a solid mass of metal without the production of eddy-currents within it; hence if the width of the slot-opening is equal to or exceeds twice the length of the single air-gap, lamination of the pole-pieces in the same plane as that of the armature core becomes advisable.
If the wires are threaded through holes or tunnels pierced close to the periphery of the core, the same advantages are gained as with open slots, and lamination of the pole-pieces is rendered unnecessary. But on the other hand, the process of winding becomes laborious and expensive, while the increase in the inductance of the coils owing to their being surrounded by a closed iron circuit is prejudicial to sparkless commutation in the continuous-current dynamo and to the regulation of the voltage of the alternator. A compromise is found in the half-closed slot, which is not uncommon in alternators, although the open slot is more usual in continuous-current dynamos.
With the addition of more turns to the elementary drum loop or of several complete coils, new questions arise, and in connexion therewith the two great classes of machines, viz. alternators and continuous-current dynamos, which have above been treated side by side, diverge considerably, so that they are best considered separately. The electromotive-force equation of the alternator will be first deduced, and subsequently that of the continuous-current machine.
Corresponding to the number of pairs of poles in the multipolar alternator, it is evident that there may also be an equal number of coils as shown diagrammatically in fig. 11. The additional coils, being similarly situated in respect to other pairs of poles, will exactly reproduce the E.M.F. of the original coil in phase and magnitude, so that when they are connected in series the total E.M.F. will be proportional to the number of coils in series; or if they are connected in parallel, while not adding to the E.M.F., they will proportionately increase the current-carrying capacity of the combination. But within each coil the addition of more loops will not cause an equal increase in the total E.M.F., unless the phases of the component E.M.F.'s due to the several turns are identical, and on this account it becomes necessary to consider the effect of the width of the coil-side.
If the additional loops are wound within the same slots as the original loop, the winding is "concentrated," and each turn will then add the same E.M.F. But if the coil-side is divided between two or more slots, the phase of the E.M.F. yielded by the wires in one slot being different from that of the wires in another neighbouring slot, the sum of all the E.M.F.'s will be less than the E.M.F. of one component loop multiplied by the number of loops or turns in the coil. The percentage reduction in the E.M.F. will depend upon the number of the slots in a coil-side and their distance apart, i.e. on the virtual width of the coil-side expressed as a fraction of the "pole-pitch" or the distance measured along the pitch-line from the centre of one pole to the centre of a neighbouring pole of opposite sign (fig. 12). The winding is now to be regarded as "grouped," since a small number of distinct phases corresponding to the groups within the two, three or four slots have to be compounded together. As the number of slots per coil-side is increased, an approach is gradually made to the case of "uniform distribution," such as would obtain in a smooth-core armature in which the turns of the coil are wound closely side by side. Thus in the six-turn coil of fig. 12 A, which represents the development of a two-pole armature when the core is cut down to the shaft and opened out flat, there are in effect six phases compounded together, each of which differs but little from that of its next neighbour. With numerous wires lying still closer together a large number of phases are compounded until the distribution becomes practically uniform; the decrease in the E.M.F., as compared with that of a single turn multiplied by the number in series, is then immediately dependent upon the width of the coil-side relatively to the pole-pitch.
If the width of the inner loop of fig. 12 A is less than that of the pole-face, its two sides will for some portion of each period be moving under the same pole, and "differential action" results, the net E.M.F. being only that due to the difference between the E.M.F.'s of the two sides. The loop of smallest width must therefore exceed the width of pole-face, if direct differential action is to be avoided. The same consideration also determines the width of the outer loop; if this be deducted from twice the pole-pitch, the difference should not be less than the width of the pole-face, so that, e.g., in a bipolar machine the outer loop may stand to the S. pole exactly as the inner loop stands to the N. pole (fig. 13). In other words, the width of the coil-side must not exceed the width of the interpolar gap between two fields. Evidently then if the ratio of the pole-width to the pole-pitch approaches unity, the width of the coil-side must be very small, and vice versa. A compromise between these conflicting considerations is found if the pole is made not much more than half the pole-pitch, and the width of the coil-side is similarly about half the pole-pitch and therefore equal in width to the pole (fig. 13). A single large coil, such as that of fig. 12 A, can, however, equally well be divided into two halves by taking the end-connexions of one half of the turns round the opposite side of the shaft (fig. 12 B), as indeed has already been done in fig. 13. Each sheaf or band of active wires corresponding to a pole is thereby unaffected, but the advantages are gained that the axial length of the end-connexions is halved, and that they have less inductance. Thus if in fig. 11 there are four turns per coil, fig. 14 is electrically equivalent to it (save that the coils are here shown divided into two parallel paths, each carrying half the total current). When the large coils are divided as above described, it results that there are as many coils as there are poles, the outer loop of the small coil having a width equal to the pole-pitch, and the inner a width equal to the pole-face.
Such is the form which the "single-phase alternator" takes, but since only one-half of the armature core is now covered with winding, an entirely distinct but similar set of coils may be wound to form a second armature circuit between the coils of the first circuit. The phase of this second circuit will differ by 90 deg. or a quarter of a period from that of the first, and it may either be used to feed an entirely separate external circuit possibly at a different pressure or, if it be composed of the same number of turns and therefore gives the same voltage, it may be interconnected with the first circuit to form a "quarter-phase alternator," as will be more fully described later. By an extension of the same process, if the width of each side of a coil is reduced to one-sixth of the pole-pitch, three armature circuits can be wound on the same core, and a "three-phase alternator," giving waves of E.M.F. differing in phase by 120 deg., is obtained.
The fundamental "electromotive-force equation" of the heteropolar alternator can now be given a more definite form. Let Z_a be the number of C. G. S. lines or the total flux, which issuing from any one pole flows through the armature core, to leave it by another pole of opposite sign. Since each active wire cuts these lines, first as they enter the armature core and then as they emerge from it to enter another pole, the total number of lines cut in one revolution by any one active wire is 2pZ_a. The time in seconds taken by one revolution is 60/N. The average E.M.F. induced in each active wire in one revolution being proportional to the number of lines cut divided by the time taken to cut them is therefore 2Z_a(pN/60) x 10^(-8) volts. The active wires which are in series and form one distinct phase may be divided into as many bands as there are poles; let each such band contain t active wires, which as before explained may either form one side of a single large coil or the adjacent sides of two coils when the large coil is divided into two halves. Since the wires are joined up into loops, two bands are best considered together, which with either arrangement yield in effect a single coil of t turns. The average E.M.F.'s of all the wires in the two bands when added together will therefore be 4Z_a(pN/60)t x 10^(-8). But unless each band is concentrated within a single slot, there must be some differential action as they cross the neutral line between the poles, so that the last expression is virtually the _gross_ average E.M.F. of the loops on the assumption that the component E.M.F.'s always act in agreement round the coil and do not at times partially neutralize one another. The _net_ average E.M.F. of the coil as a whole, or the arithmetical mean of all the instantaneous values of a half-wave of the actual E.M.F. curve, is therefore reduced to an extent depending upon the amount of differential action and so upon the width of the coil-side when this is not concentrated. Let k' = the coefficient by which the gross average E.M.F. must be multiplied to give the net average E.M.F.; then k' may be called the "width-factor," and will have some value less than unity when the wires of each band are spread over a number of slots. The net average E.M.F. of the two bands corresponding to a pair of poles is thus e_(av) = 4k'Z_a(pN/60)t x 10^(-8).
The shape of the curve of instantaneous E.M.F. of the coil must further be taken into account. The "effective" value of an alternating E.M.F. is equal to the square root of the mean square of its instantaneous values, since this is the value of the equivalent unidirectional and unvarying E.M.F., which when applied to a given resistance develops energy at the same rate as the alternating E.M.F., when the effect of the latter is averaged over one or any whole number of periods. Let k" = the ratio of the square root of the mean square to the average E.M.F. of the coil, i.e. = effective E.M.F./average E.M.F. Since it depends upon the shape of the E.M.F. curve, k" is also known as the "form-factor"; thus if the length of gap between pole-face and armature core and the spacing of the wires were so graduated as to give a curve of E.M.F. varying after a sine law, the form-factor would have the particular value of [pi]/2[root]2 = 1.11, and to this condition practical alternators more or less conform. The effective E.M.F. of the two bands corresponding to a pair of poles is thus e_(eff) = 4k'k"Z_a(pN/60)t x 10^(-8).
In any one phase there are p pairs of bands, and these may be divided into q parallel paths, where q is one or any whole number of which p is a multiple. The effective E.M.F. of a complete phase is therefore pe{eff}/q. Lastly, if m = the number of phases into which the armature winding is divided, and [tau] = the total number of active wires on the armature counted all round its periphery, t = [tau]/2pm, and the effective E.M.F. per phase is E_a = 2k'k"Z_a(pN[tau]/60mq) x 10^(-8).
The two factors k' and k" may be united into one coefficient, and the equation then takes its final form
E_a = 2KZ_a(pN[tau]/60mq) x 10^(-8) volts (1a)
In the alternator q is most commonly 1, and there is only one circuit per phase; finally the value of K or the product of the width-factor and the form-factor usually falls between the limits of 1 and 1.25.
We have next to consider the effect of the addition of more armature loops in the case of dynamos which give a unidirectional E.M.F. in virtue of their split-ring collecting device, i.e. of the type shown in fig. 7 with drum armature or its equivalent ring form. As before, if the additional loops are wound in continuation of the first as one coil connected to a single split-ring, this coil must be more or less concentrated into a narrow band; since if the width becomes nearly equal to or exceeds the width of the interpolar gap, the two edges of the coil-side will just as in the alternator act differentially against one another during part of each revolution. The drum winding with a single coil thus gives an armature of the H- or "shuttle" form invented by Dr Werner von Siemens. Although the E.M.F. of such an arrangement may have a much higher maximum value than that of the curve of fig. 7 for a single loop, yet it still periodically varies during each revolution and so gives a pulsating current, which is for most practical uses unsuitable. But such pulsation might be largely reduced if, for example, a second coil were placed at right angles to the original coil and the two were connected in series; the crests of the wave of E.M.F. of the second coil will then coincide with the hollows of the first wave, and although the maximum of the resultant curve of E.M.F. may be no higher its fluctuations will be greatly decreased. A spacial displacement of the new coils along the pole-pitch, somewhat as in a polyphase machine, thus suggests itself, and the process may be carried still further by increasing the number of equally spaced coils, provided that they can be connected in series and yet can have their connexion with the external circuit reversed as they pass the neutral line between the poles.
Given two coils at right angles and with their split-rings displaced through a corresponding angle of 90 deg., they may be connected in series by joining one brush to the opposite brush of the second coil, the external circuit being applied to the two remaining brushes.[14] The same arrangement may again be repeated with another pair of coils in parallel with the first, and we thus obtain fig. 15 with four split-rings, their connexions to the loops being marked by corresponding numerals; the four coils will give the same E.M.F. as the two, but they will be jointly capable of carrying twice the current, owing to their division into two parallel circuits. Now in place of the four split-rings may be employed the greatly simplified four-segment structure shown in fig. 16, which serves precisely the same purpose as the four split-rings but only requires two instead of eight brushes. The effect of joining brush 2 in fig. 15 across to brush 3, brush 4 to brush 5, 5 to 6, &c., has virtually been to connect the end of coil A with the beginning of coil B, and the end of coil B with the beginning of coil A', and so on, until they form a continuous closed helix. Each sector of fig. 16 will therefore replace two halves of a pair of adjacent split-rings, if the end and beginning of a pair of adjacent coils are connected to it in a regular order of sequence. The four sectors are insulated from one another and from the shaft, and the whole structure is known as the "commutator,"[15] its function being not simply to collect the current but also to commute its direction in any coil as it passes the interpolar gap. The principle of the "closed-coil continuous-current armature" is thus reached, in which there are at least two parallel circuits from brush to brush, and from which a practically steady current can be obtained. Each coil is successively short-circuited, as a brush bridges over the insulation between the two sectors which terminate it; and the brushes must be so set that the period of short-circuit takes place when the coil is generating little or no E.M.F., i.e. when it is moving through the zone between the pole-tips. The effect of the four coils in reducing the percentage fluctuation of the E.M.F. is very marked, as shown at the foot of fig. 15 (where the upper curve is the resultant obtained by adding together the separate curves of coils A and B), and the levelling process may evidently be carried still further by the insertion of more coils and more corresponding sectors in the commutator, until the whole armature is covered with winding. For example, figs. 17 and 18 show a ring and a drum armature, each with eight coils and eight commutator sectors; their resultant curve, on the assumption that a single active wire gives the flat-topped curve of fig. 4, will be the upper wavy line of E.M.F. obtained by adding together two of the resultant curves of fig. 15, with a relative displacement of 45 deg.. The amount of fluctuation for a given number of commutator sectors depends upon the shape of the curve of E.M.F. yielded by the separate small sections of the armature winding; the greater the polar arc, the less the fluctuation. In practice, with a polar arc equal to about 0.75 of the pitch, any number of sectors over 32 per pair of poles yields an E.M.F. which is sensibly constant throughout one or any number of revolutions.
The fundamental electro-motive-force equation of the continuous-current heteropolar machine is easily obtained by analogy from that of the alternator. The gross average E.W.F. from the two sides of a drum loop without reference to its direction is as before 4Z_a(pN/60) x 10^(-8) volts. But for two reasons its net average E.M.F. may be less; the span of the loop may be less than the pole-pitch, so that even when the brushes are so set that the position of short-circuit falls on the line where the field changes its direction, the two sides of the loop for some little time act against each other; or, secondly, even if the span of the loop be equal to the pole-pitch, the brushes may be so set that the reversal of the direction of its induced E.M.F. does not coincide with reversal of the current by the passage of the coil under the brushes. The net average E.M.F. of the loop is therefore proportional to the algebraic sum of the lines which it cuts in passing from one brush to another, and this is equal to the net amount of the flux which is included within the loop when situated in the position of short-circuit under a brush. The amount of this flux may be expressed as k'Z_a where k' is some coefficient, less than unity if the span of the coil be less than the pole-pitch, and also varying with the position of the brushes. The net average E.M.F. of the loop is therefore
4k'Z_a(pN/60) x 10^(-8).
In practice the number of sections of the armature winding is so large and their distribution round the armature periphery is so uniform, that the sum total of the instantaneous E.M.F.'s of the several sections which are in series becomes at any moment equal to the net average E.M.F. of one loop multiplied by the number which are in series. If the winding is divided into q parallel circuits, the number of loops in series is [tau]/2q, so that the total E.M.F. is E_a = 2(k'/q)Z_a(pN/60)[tau] x 10^(-8) volts. Thus as compared with the alternator not only is there no division of the winding into separate phases, but the form-factor k' disappears, since the effective and average E.M.F.'s are the same. Further whereas in the alternator q may = 1, in the continuous-current closed-coil armature there can never be less than two circuits in parallel from brush to brush, and if more, their number must always be a multiple of two, so that q can never be less than two and must always be an even number. Lastly, the factor k' is usually so closely equal to 1, that the simplified equation may in practice be adopted, viz.
E_a = (2/q)(ZpN/60)[tau] x 10^(-8) volts (1b)
The fundamental equation of the electromotive force of the dynamo in its fully developed forms (1 a) (and 1 b) may be compared with its previous simple statement (1.). The three variable terms still find their equivalents, but are differently expressed, the density B_g being replaced by the total flux of one field Z_a, the length L of the single active wire by the total number of such wires [tau], and the velocity of movement V by the number of revolutions per second. Even when the speed is fixed, an endless number of changes may be rung by altering the relative values of the remaining two factors; and in successful practice these may be varied between fairly wide limits without detriment to the working or economy of the machine. While it may be said that the equation of the E.M.F. was implicitly known from Faraday's time onwards, the difficulty under which designers laboured in early days was the problem of choosing the correct relation of Z_a or [tau] for the required output; this, again, was due chiefly to the difficulty of predetermining the total flux before the machine was constructed. The general error lay in employing too weak a field and too many turns on the armature, and credit must here be given to the American inventors, E. Weston and T.A. Edison, for their early appreciation of the superiority in practical working of the drum armature, with comparatively few active wires rotating in a strong field.
The armature core.
_Continuous-current Dynamos._--On passing to the separate consideration of alternators and continuous-current dynamos, the chief constructive features of the latter will first be taken in greater detail. As already stated in the continuous-current dynamo the armature is usually the rotating portion, and the necessity of laminating its core has been generally described. The thin iron stampings employed to build up the core take the form of circular washers or "disks," which in small machines are strung directly on the shaft; in larger multipolar machines, in which the required radial depth of iron is small relatively to the diameter, a central cast iron hub supports the disks. Since the driving force is transmitted through the shaft to the disks, they must in the former case be securely fixed by keys sunk into the shaft; when a central hub is employed (fig. 19) it is keyed to the shaft, and its projecting arms engage in notches stamped on the inner circumference of the disks, or the latter have dovetailed projections fitting into the arms. The disks are then tightly compressed and clamped between stout end-plates so as to form a nearly solid iron cylinder of axial length slightly exceeding the corresponding dimension of the poles. If the armature is more than 4 ft. in diameter, the disks become too large to be conveniently handled in one piece, and are therefore made in segments, which are built up so as to break joint alternately. Prior to assemblage, the external circumference of each disk is notched in a stamping machine with the required number of slots to receive the armature coils, and the longitudinal grooves thereby formed in the finished core only require to have their sharp edges smoothed off so that there may be no risk of injury to the insulation of the coils.
Armature winding.
With open slots either the armature coils may be encased with wrappings of oiled linen, varnished paper and thin flexible micanite sheeting in order to insulate them electrically from the iron slots in which they are afterwards embedded; or the slots may be themselves lined with moulded troughs of micanite, &c., for the reception of the armature coils, the latter method being necessary with half-closed slots. According to the nature of the coils armatures may be divided into the two classes of coil-wound and bar-wound. In the former class, round copper wire, double-cotton covered, is employed, and the coils are either wound by hand directly on to the armature core, or are shaped on formers prior to being inserted in the armature slots. Hand-winding is now only employed in very small bipolar machines, the process being expensive and accompanied by the disadvantage that if one section requires to be repaired, the whole armature usually has to be dismantled and re-wound. Former-wound coils are, on the other hand, economical in labour, perfectly symmetrical and interchangeable, and can be thoroughly insulated before they are placed in the slots. The shapers employed in the forming process are very various, but are usually arranged to give to the finished coil a lozenge shape, the two straight active sides which fit into the straight slots being joined by V-shaped ends; at each apex of the coil the wire is given a twist, so that the two sides fall into different levels, an upper and a lower, corresponding to the two layers which the coil-sides form on the finished armature. Rectangular wire of comparatively small section may be similarly treated, and if only one loop is required per section, wide and thin strip can be bent into a complete loop, so that the only soldered joints are those at the commutator end where the loops are interconnected. But finally with massive rectangular conductors, the transition must be made to bar-winding, in which each bar is a half-loop, insulated by being taped after it has been bent to the required shape; the separate bars are arranged on the armature in two layers, and their ends are soldered together subsequently to form loops. As a general rule, whether bars or former-wound coils are employed, the armature is barrel-wound, i.e. the end-connexions project outwards from the slots with but little change of level, so that they form a cylindrical mass supported on projections from the end-plates of the core (fig. 19); but, in certain cases, the end-connexions are bent downwards at right angles to the shaft, and they may then consist of separate strips of copper bent to a so-called butterfly or evolute shape.
After the coils or loops have been assembled in the slots on the armature core, and the commutator has been fixed in place on the shaft, the soldering of the ends of the coils proceeds, by which at once the union of the end of one coil with the beginning of the next, and also their connexion to the commutator sectors, is effected, and in this lies the essential part of armature winding.
Lap-winding.
The development of the modern drum armature, with its numerous coils connected in orderly sequence into a symmetrical winding, as contrasted with the earlier Siemens armatures, was initiated by F. von Hefner Alteneck (1871), and the laws governing the interconnexion of the coils have now been elaborated into a definite system of winding formulae. Whatever the number of wires or bars in each side of a coil, i.e. whether it consist of a single loop or of many turns, the final connexions of its free ends are not thereby affected, and it may be mentally replaced by a single loop with two active inducing sides. The coil-sides in their final position are thus to be regarded as separate primary elements, even in number, and distributed uniformly round the armature periphery or divided into small, equally spaced groups by being located within the slots of a toothed armature. Attention must then be directed simply to the span of the back connexion between the elements at the end of the armature further from the commutator, and to the span of the front connexion by which the last turn of a coil is finally connected to the first turn of the next in sequence, precisely as if each coil of many turns were reduced to a single loop. In order to avoid direct differential action, the span of the back connexion which fixes the width of the coil must exceed the width of the pole-face, and should not be far different from the pole-pitch; it is usually a little less than the pole-pitch. Taking any one element as No. 1 in fig. 20, where for simplicity a smooth-core bipolar armature is shown, the number of winding-spaces, each to be occupied by an element, which must be counted off in order to find the position of the next element in series, is called the "pitch" of the end-connexion, front or back, as the case may be. Thus the back pitch of the winding as marked by the dotted line in fig. 20 is 7, the second side of the first loop being the element numbered 1 + 7 = 8. In forming the front end-connexion which completes the loop and joins it to the next in succession, two possible cases present themselves. By the first, or "lap-winding," the front end-connexion is brought backwards, and passing on its way to a junction with a commutator sector is led to a third element lying within the two sides of the first loop, i.e. the second loop starts with the element, No. 3, lying next but one to the starting-point of the first loop. The winding therefore returns backwards on itself to form each front end, but as a whole it works continually forwards round the armature, until it finally "re-enters," after every element has been traversed. The development of the completed winding on a flat surface shows that it takes the form of a number of partially overlapping loops, whence its name originates. The firm-line portion of fig. 21 gives the development of an armature similar to that of fig. 18 when cut through at the point marked X and opened out; two of the overlapping loops are marked thereon in heavy lines. The multipolar lap-wound armature is obtained by simply repeating the bipolar winding p times, as indicated by the dotted additions of fig. 21 which convert it from a two-pole to a four-pole machine. The characteristic feature of the lap-wound armature is that there are as many parallel paths from brush to brush, and as many points at which the current must be collected, as there are poles. As the bipolar closed-coil continuous-current armature has been shown to consist in reality of two circuits in parallel, each giving the same E.M.F. and carrying half the total current, so the multipolar lap-wound drum consists of p pairs of parallel paths, each giving the same E.M.F. and carrying 1/2p of the total current. Thus in equation 1.b we have q = 2p, and the special form which the _E.M.F. equation of the lap-wound armature_ takes is E_[alpha] = Z_a (N/60)[tau] x 10^(-8) volts. All the brushes which are of the same sign must be connected together in order to collect the total armature current. The several brush-sets of the multipolar lap-wound machine may again be reduced to two by "cross-connexion" of sectors situated 360 deg./p apart, but this is seldom done, since the commutator must then be lengthened p times in order to obtain the necessary brush contact-surface for the collection of the entire current.
Wave-winding.
But for many purposes, especially where the voltage is high and the current small, it is advantageous to add together the inductive effect of the several poles of the multipolar machine by throwing the E.M.F's of half the total number of elements into series, the number of parallel circuits being conversely again reduced to two. This is effected by the second method of winding the closed-coil continuous current drum, which is known as "wave-winding." The front pitch is now in the same direction round the armature as the back pitch (fig. 22), so that the beginning of the second loop, i.e. element No. 15, lies outside the first loop. After p loops have been formed and as many elements have been traversed as there are poles, the distance covered either falls short of or exceeds a complete tour of the armature by two winding-spaces, or the width of two elements. A second and third tour are then made, and so on, until finally the winding again closes upon itself. When the completed winding is developed as in fig. 23, it is seen to work continuously forwards round the armature in zigzag waves, one of which is marked in heavy lines, and the number of complete tours is equal to the average of the back and front pitches. Since the number of parallel circuits from brush to brush is q = 2, the _E.M.F. equation of the wave-wound drum_ is E_a = pZ_a (N/60)[tau] x 10^(-8) volts. Only two sets of brushes are necessary, but in order to shorten the length of the commutator, other sets may also be added at the point of highest and lowest potential up to as many in number as there are poles. Thus the advantage of the wave-wound armature is that for a given voltage and number of poles the number of active wires is only 1/p of that in the lap-wound drum, each being of larger cross-section in order to carry p times as much current; hence the ratio of the room occupied by the insulation to the copper area is less, and the available space is better utilized. A further advantage is that the two circuits from brush to brush consist of elements influenced by all the poles, so that if for any reason, such as eccentricity of the armature within the bore of the pole-pieces, or want of uniformity in the magnetic qualities of the poles, the flux of each field is not equal to that of every other, the equality of the voltage produced by the two halves of the winding is not affected thereby.
In appearance the two classes of armatures, lap and wave, may be distinguished in the barrel type of winding by the slope of the upper layer of back end-connexions, and that of the front connexions at the commutator end being parallel to one another in the latter, and oppositely directed in the former.
After completion of the winding, the end-connexions are firmly bound down by bands of steel or phosphor bronze binding wire, so as to resist the stress of centrifugal force. In the case of smooth-surface armatures, such bands are also placed at intervals along the length of the armature core, but in toothed armatures, although the coils are often in small machines secured in the slots by similar bands of a non-magnetic high-resistance wire, the use of hard-wood wedges driven into notches at the sides of the slots becomes preferable, and in very large machines indispensable. The external appearance of a typical armature with lap-winding is shown in fig. 24.
The commutator.
A sound mechanical construction of the commutator is of vital importance to the good working of the continuous-current dynamo. The narrow, wedge-shaped sectors of hard-drawn copper, with their insulating strips of thin mica, are built up into a cylinder, tightly clamped together, and turned in the lathe; at each end a V-shaped groove is turned, and into these are fitted rings of micanite of corresponding section (fig. 19); the whole is then slipped over a cast iron sleeve, and at either end strong rings are forced into the V-shaped grooves under great pressure and fixed by a number of closely-pitched tightening bolts. In dynamos driven by steam-turbines in which the peripheral speed of the commutator is very high, rings of steel are frequently shrunk on the surface of the commutator at either end and at its centre. But in every case the copper must be entirely insulated from the supporting body of metal by the interposition of mica or micanite and the prevention of any movement of the sectors under frequent and long-continued heating and cooling calls for the greatest care in both the design and the manufacture.
Forms of field-magnet.
On passing to the second fundamental part of the dynamo, namely, the field-magnet, its functions may be briefly recalled as follows:--It has to supply the magnetic flux; to provide for it an iron path as nearly closed as possible upon the armature, save for the air-gaps which must exist between the pole-system and the armature core, the one stationary and the other rotating; and, lastly, it has to give the lines such direction and intensity within the air-gaps that they may be cut by the armature wires to the best advantage. Roughly corresponding to the three functions above summarized are the three portions which are more or less differentiated in the complete structure. These are: (1) the magnet "cores" or "_limbs_," carrying the exciting coils whereby the inert iron is converted into an electro-magnet; (2) the _yoke_, which joins the limbs together and conducts the flux between them; and (3) the _pole-pieces_, which face the armature and transmit the lines from the limbs through the air-gap to the armature core, or vice versa.
Of the countless shapes which the field-magnet may take, it may be said, without much exaggeration, that almost all have been tried; yet those which have proved economical and successful, and hence have met with general adoption, may be classed under a comparatively small number of types. For bipolar machines the _single horse-shoe_ (fig. 25), which is the lineal successor of the permanent magnet employed in the first magneto-electric machines, was formerly very largely used. It takes two principal forms, according as the pole-pieces and armature are above or beneath the magnet limbs and yoke. The "over-type" form is best suited to small belt-driven dynamos, while the "under-type" is admirably adapted to be directly driven by the steam-engine, the armature shaft being immediately coupled to the crank-shaft of the engine. In the latter case the magnet must be mounted on non-magnetic supports of gun-metal or zinc, so as to hold it at some distance away from the iron bedplate which carries both engine and dynamo; otherwise a large proportion of the flux which passes through the magnet limbs would leak through the bedplate across from pole to pole without passing through the armature core, and so would not be cut by the armature wires.
Next may be placed the "Manchester" field (fig. 26)--the type of a divided magnetic circuit in which the flux forming one field or pole is divided between two magnets. An exciting coil is placed on each half of the double horse-shoe magnet, the pair being so wound that consequent poles are formed above and below the armature. Each magnet thus carries one-half of the total flux, the lines of the two halves uniting to form a common field where they issue forth into or leave the air-gaps. The pole-pieces may be lighter than in the single horse-shoe type, and the field is much more symmetrical, whence it is well suited to ring armatures of large diameter. Yet these advantages are greatly discounted by the excessive magnetic leakage, and by the increased weight of copper in the exciting coils. Even if the greater percentage which the leakage lines bear to the useful flux is neglected, and the cross sectional area of each magnet core is but half that of the equivalent single horse-shoe, the weight of wire in the double magnet for the same rise of temperature in the coils must be some 40% more than in the single horse-shoe, and the rate at which energy is expended in heating the coils will exceed that of the single horse-shoe in the same proportion.
Thirdly comes the two-pole _ironclad_ type, so called from the exciting coil being more or less encased by the iron yoke; this latter is divided into two halves, which pass on either side of the armature. Unless the yoke be kept well away from the polar edges and armature, the leakage across the air into the yoke becomes considerable, especially if only one exciting coil is used, as in fig. 27 A; it is better, therefore, to divide the excitation between two coils, as in fig. 27 B, when the field also becomes symmetrical.
From this form is easily derived the _multipolar_ type of fig. 28 or fig. 29, which is by far the most usual for any number of poles from four upwards; its leakage coefficient is but small, and it is economical in weight both of iron and copper.
Materials of magnets.
As regards the materials of which magnets are made, generally speaking there is little difference in the permeability of "wrought iron" or "mild steel forgings" and good "cast steel"; typical (B, H) curves connecting the magnetizing force required with different flux-densities for these materials are given under ELECTROMAGNETISM. On the other hand there is a marked inferiority in the case of "cast iron," which for a flux-density of B = 8000 C.G.S. lines per sq. cm. requires practically the same number of ampere-turns per centimetre length as steel requires for B = 16,000. Whatever the material, if the flux-density be pressed to a high value the ampere-turns are very largely increased owing to its approaching saturation, and this implies either a large amount of copper in the field coils or an undue expenditure of electrical energy in their excitation. Hence there is a limit imposed by practical considerations to the density at which the magnet should be worked, and this limit may be placed at about B = 16,000 for wrought iron or steel, and at half this value for cast iron. For a given flux, therefore, the cast iron magnet must have twice the sectional area and be twice as heavy, although this disadvantage is partly compensated by its greater cheapness. If, however, cast iron be used for the portion of the magnetic circuit which is covered with the exciting coils, the further disadvantage must be added that the weight of copper on the field-magnet is much increased, so that it is usual to employ forgings or cast steel for the magnet cores on which the coils are wound. If weight is not a disadvantage, a cast iron yoke may be combined with the wrought iron or cast steel magnet cores. An absence of joints in the magnetic circuit is only desirable from the point of view of economy of expense in machining the component parts during manufacture; when the surfaces which abut against each other are drawn firmly together by screws, the want of homogeneity at the joint, which virtually amounts to the presence of a very thin film of air, produces little or no effect on the total reluctance by comparison with the very much longer air-gaps surrounding the armature. In order to reduce the eddy-currents in the pole-pieces, due to the use of toothed armatures with relatively wide slots, the poles themselves must be laminated, or must have fixed to them laminated pole-shoes, built up of thin strips of mild steel riveted together (as shown in fig. 29).
However it be built up, the mechanical strength of the magnet system must be carefully considered. Any two surfaces between which there exists a field of density B_g experience a force tending to draw them together proportional to the square of the density, and having a value of B_g squared/(1.735 x 10^6) lb. per sq. in. of surface, over which the density may be regarded as having the uniform value B_g. Hence, quite apart from the torque with which the stationary part of the dynamo tends to turn with the rotating part as soon as current is taken out of the armature, there exists a force tending to make the pole-pieces close on the armature as soon as the field is excited. Since both armature and magnet must be capable of resisting this force, they require to be rigidly held; although the one or the other must be capable of rotation, there should otherwise be no possibility of one part of the magnetic circuit shifting relatively to any other part. An important conclusion may be drawn from this circumstance. If the armature be placed exactly concentric within the bore of the poles, and the two or more magnetic fields be symmetrical about a line joining their centres, there is no tendency for the armature core to be drawn in one direction more than in another; but if there is any difference between the densities of the several fields, it will cause an unbalanced stress on the armature and its shaft, under which it will bend, and as this bending is continually reversed relatively to the fibres of the shaft, they will eventually become weakened and give way. Especially is this likely to take place in dynamos with short air-gaps, wherein any difference in the lengths of the air-gaps produces a much greater percentage difference in the flux-density than in dynamos with long air-gaps. In toothed armatures with short air-gaps the shaft must on this account be sufficiently strong to withstand the stress without appreciable bending.
The magnetic circuit.
Reference has already been made to the importance in dynamo design of the _predetermination of the flux_ due to a given number of ampere-turns wound on the field-magnet, or, conversely, of the number of ampere-turns which must be furnished by the exciting coils in order that a certain flux corresponding to one field may flow through the armature core from each pole. An equally important problem is the correct proportioning of the field-magnet, so that the useful flux Z_a may be obtained with the greatest economy in materials and exciting energy. The key to the two problems is to be found in the concept of a magnetic circuit as originated by H.A. Rowland and R.H.M. Bosanquet;[16] and the full solution of both may be especially connected with the name of Dr J. Hopkinson, from his practical application of the concept in his design of the Edison-Hopkinson machine, and in his paper on "Dynamo-Electric Machinery."[17] The publication of this paper in 1886 begins the second era in the history of the dynamo; it at once raised its design from the level of empirical rules-of-thumb to a science, and is thus worthy to be ranked as the necessary supplement of the original discoveries of Faraday. The process of predetermining the necessary ampere-turns is described in a simple case under ELECTROMAGNETISM. In its extension to the complete dynamo, it consists merely in the division of the magnetic circuit into such portions as have the same sectional area and permeability and carry approximately the same total flux; the difference of magnetic potential that must exist between the ends of each section of the magnet in order that the flux may pass through it is then calculated _seriatim_ for the several portions into which the magnetic circuit is divided, and the separate items are summed up into one magnetomotive force that must be furnished by the exciting coils.
The chief sections of the magnetic circuit are (1) the air-gaps, (2) the armature core, and (3) the iron magnet.
The _air-gap_ of a dynamo with smooth-core armature is partly filled with copper and partly with the cotton, mica, or other materials used to insulate the core and wires; all these substances are, however, sensibly non-magnetic, so that the whole interferric gap between the iron of the pole-pieces and the iron of the armature may be treated as an air-space, of which the permeability is constant for all values of the flux density, and in the C.G.S. system is unity. Hence if l_g and A_g be the length and area of the single air-gap in cm. and sq. cm., the reluctance of the double air-gap is 2l_g/A_g, and the difference of magnetic potential required to pass Z_a lines over this reluctance is Z_a.2l_g/A_g = B_g.2l_g; or, since one ampere-turn gives 1.257 C.G.S. units of magnetomotive force, the exciting power in ampere-turns required over the two air-gaps is X_g = B_g.2l_g/1.257 = 0.8 B_g.2l_g. In the determination of the area A_g small allowance must be made for the fringe of lines which extend beyond the actual polar face. In the toothed armature with open slots, the lines are no longer uniformly distributed over the air-gap area, but are graduated into alternate bands of dense and weak induction corresponding to the teeth and slots. Further, the lines curve round into the sides of the teeth, so that their average length of path in the air and the air-gap reluctance is not so easily calculated. Allowance must be made for this by taking an increased length of air-gap = ml_g, where m is the ratio _maximum density/mean density_, of which the value is chiefly determined by the ratios of the width of tooth to width of slot and of the width of slot to the air-gap between pole-face and surface of the armature core.
The _armature core_ must be divided into the teeth and the core proper below the teeth. Owing to the tapering section of the teeth, the density rises towards their root, and when this reaches a high value, such as 18,000 or more lines per sq. cm., the saturation of the iron again forces an increasing proportion of the lines outwards into the slot. A distinction must then be drawn between the "apparent" induction which would hold if all the lines were concentrated in the teeth, and the "real" induction. The area of the iron is obtained by multiplying the number of teeth under the pole-face by their width and by the net length of the iron core parallel to the axis of rotation. The latter is the gross length of the armature less the space lost through the insulating varnish or paper between the disks or through the presence of ventilating ducts, which are introduced at intervals along the length of the core. The former deduction averages about 7 to 10% of the gross length, while the latter, especially in large multipolar machines, is an even more important item. Alter calculating the density at different sections of the teeth, reference has now to be made to a (B, H) or flux-density curve, from which may be found the number of ampere-turns required per cm. length of path. This number may be expressed as a function of the density in the teeth, and f(B_t) be its average value over the length of a tooth, the ampere-turns of excitation required over the teeth on either side of the core as the lines of one field enter or leave the armature is X_t = f(B_t).2l_t, where l_t is the length of a single tooth in cm.
In the core proper below the teeth the length of path continually shortens as we pass from the middle of the pole towards the centre line of symmetry. On the other hand, as the lines gradually accumulate in the core, their density increases from zero midway under the poles until it reaches a maximum on the line of symmetry. The two effects partially counteract one another, and tend to equalize the difference of magnetic potential required over the paths of varying lengths; but since the reluctivity of the iron increases more rapidly than the density of the lines, we may approximately take for the length of path (l_a) the minimum peripheral distance between the edges of adjacent pole-faces, and then assume the maximum value of the density of the lines as holding throughout this entire path. In ring and drum machines the flux issuing from one pole divides into two halves in the armature core, so that the maximum density of lines in the armature is B_a = Z_a/2ab, where a = the radial depth of the disks in centimetres and b = the net length of iron core. The total exciting power required between the pole-pieces is therefore, at no load, X_p = X_g + X_t + X_a, where X_a = f(B_a).l_a; in order, however, to allow for the effect of the armature current, which increases with the load, a further term X_b, must be added.
In the continuous-current dynamo it may be, and usually is, necessary to move the brushes forward from the interpolar line of symmetry through a small angle in the direction of rotation, in order to avoid sparking between the brushes and the commutator (_vide infra_). When the dynamo is giving current, the wires on either side of the diameter of commutation form a current-sheet flowing along the surface of the armature from end to end, and whatever the actual end-connexions of the wires, the wires may be imagined to be joined together into a system of loops such that the two sides of each loop are carrying current in opposite directions. Thus a number of armature ampere-turns are formed, and their effect on the entire system of magnet and armature must be taken into account. So long as the diameter of commutation coincides with the line of symmetry, the armature may be regarded as a cylindrical electromagnet producing a flux of lines, as shown in fig. 30. The direction of the self-induced flux in the air-gaps is the same as that of the lines of the external field in one quadrant on one side of DC, but opposed to it in the other quadrant on the same side of DC; hence in the resultant field due to the combined action of the field-magnet and armature ampere-turns, the flux is as much strengthened over the one half of each polar face as it is weakened over the other, and the total number of lines is unaffected, although their distribution is altered. The armature ampere-turns are then called _cross-turns_, since they produce a cross-field, which, when combined with the symmetrical field, causes the leading pole-corners ll to be weakened and the trailing pole-corners tt to be strengthened, the neutral line of zero field being thus twisted forwards in the direction of rotation. But when the brushes and diameter of commutation are shifted forward, as shown in fig. 31, it will be seen that a number of ampere-turns, forming a zone between the lines Dn and mC, are in effect wound immediately on the magnetic circuit proper, and this belt of ampere-turns is in direct opposition to the ampere-turns of the field, as shown by the dotted and crossed wires on the pole-pieces. The armature ampere-turns are then divisible into the two bands, the _back-turns_, included within twice the angle of lead [lambda], weakening the field, and the cross-turns, bounded by the lines Dm, nC, again producing distortion of the weakened symmetrical field. If, therefore, a certain flux is to be passed through the armature core in opposition to the demagnetizing turns, the difference of magnetic potential between the pole-faces must include not only X_a, X_t, and X_g, but also an item X_b, in order to balance the "back" ampere-turns of the armature. The amount by which the brushes must be shifted forward increases with the armature current, and in corresponding proportion the back ampere-turns are also increased, their value being c[tau]2[lambda]/360 deg., where c = the current carried by each of the [tau] active wires. Thus the term X_b, takes into account the effect of the armature reaction on the total flux; it varies as the armature current and angle of lead required to avoid sparking are increased; and the reason for its introduction in the fourth place (X_p = X_g + X_t + X_a + X_b), is that it increases the magnetic difference of potential which must exist between the poles of the dynamo, and to which the greater part of the leakage is due. The leakage paths which are in parallel with the armature across the poles must now be estimated, and so a new value be derived for the flux at the commencement of the _iron-magnet_ path. If P = their joint permeance, the leakage flux due to the difference of potential at the poles is z_l = 1.257X_p x P, and this must be added to the useful flux Z_a, or Z_p = Z_a + Z_l. There are also certain leakage paths in parallel with the magnet cores, and upon the permeance of these a varying number of ampere-turns is acting as we proceed along the magnet coils; the magnet flux therefore increases by the addition of leakage along the length of the limbs, and finally reaches a maximum near the yoke. Either, then, the density in the magnet B_m = Z_m/A_m will vary if the same sectional area be retained throughout, or the sectional area of the magnet must itself be progressively increased. In general, sufficient accuracy will be obtained by assuming a certain number of additional leakage lines z_n as traversing the entire length of magnet limbs and yoke (= l_m), so that the density in the magnet has the uniform value B_m = (Z_p + z_n)/A_m. The leakage flux added on actually within the length of the magnet core or z_n will be approximately equal to half the total M.M.F. of the coils multiplied by the permeance of the leakage paths around one coil. The corresponding value of H can then be obtained from the (B, H) curve of the material of which the magnet is composed, and the ampere-turns thus determined must be added to X_p, or X = X_p + X_m, where X_m = f(B_m)l_m. The final equation for the exciting power required on a magnetic circuit as a whole will therefore take the form
X = A[Tau] = 0.8B_g.2l_g + f(B_t)2l_t + f(B_a)l_a + X_b + f(B_m)l_m. (3)
If the magnet cores are of wrought iron or cast steel, and the yoke is of cast iron, the last term must be divided into two portions corresponding to the different materials, i.e. into f(B_m)l_m + f(B_y)l_y. In the ordinary multipolar machine with as many magnet-coils as there are poles, each coil must furnish half the above number of ampere-turns.
Magnetic leakage.
Since no substance is impermeable to the passage of magnetic flux, the only form of magnetic circuit free from leakage is one uniformly wound with ampere-turns over its whole length. The reduction of the _magnetic leakage_ to a minimum in any given type is therefore primarily a question of distributing the winding as far as possible uniformly upon the circuit, and as the winding must be more or less concentrated into coils, it resolves itself into the necessity of introducing as long air-paths as possible between any surfaces which are at different magnetic potentials. No iron should be brought near the machine which does not form part of the magnetic circuit proper, and especially no iron should be brought near the poles, between which the difference of magnetic potential practically reaches its maximum value. In default of a machine of the same size or similar type on which to experiment, the probable direction of the leakage flux must be assumed from the drawing, and the air surrounding the machine must be mapped out into areas, between which the permeances are calculated as closely as possible by means of such approximate formulae as those devised by Professor G. Forbes.
Excitation of field-magnet.
In the earliest "magneto-electric" machines permanent steel magnets, either simple or compound, were employed, and for many years these were retained in certain alternators, some of which are still in use for arc lighting in lighthouses. But since the field they furnish is very weak, a great advance was made when they were replaced by soft iron electromagnets, which could be made to yield a much more intense flux. As early as 1831 Faraday[18] experimented with electromagnets, and after 1850 they gradually superseded the permanent magnet. When the total ampere-turns required to excite the electromagnet have been determined, it remains to decide how the excitation shall be obtained; and, according to the method adopted, continuous-current machines may be divided into four well-defined classes.
The simplest method, and that which was first used, is _separate excitation_ from some other source of direct current, which may be either a primary or a secondary battery or another dynamo (fig. 32). But since the armature yields a continuous current, it was early suggested (by J. Brett in 1848 and F. Sinsteden in 1851) that this current might be utilized to increase the flux; combinations of permanent and electromagnets were therefore next employed, acting either on the main armature or on separate armatures, until in 1867 Dr Werner von Siemens and Sir C. Wheatstone almost simultaneously discovered that the dynamo could be made _self-exciting_ through the residual magnetism retained in the soft iron cores of the electromagnet. The former proposed to take the whole of the current round the magnet coils which were in series with the armature and external circuit, while the latter proposed to utilize only a portion derived by a shunt from the main circuit; we thus arrive at the second and third classes, namely, _series_ and _shunt_ machines. The starting of the process of excitation in either case is the same; when the brushes are touching the commutator and the armature is rotated, the small amount of flux left in the magnet is cut by the wires, and a very small current begins to flow round the closed circuit; this increases the flux, which in turn further increases the E.M.F. and current, until, finally, the cumulative effect stops through the increasing saturation of the iron cores. Fig. 33, illustrating the _series_ machine, shows the winding of the exciting coils to be composed of a few turns of thick wire. Since the current is undivided throughout the whole circuit, the resistance of both the armature and field-magnet winding must be low as compared with that of the external circuit, if the useful power available at the terminals of the machine is to form a large percentage of the total electrical power--in other words, if the efficiency is to be high. Fig. 34 shows the third method, in which the winding of the field-magnets is a _shunt_ or fine-wire circuit of many turns applied to the terminals of the machine; in this ease the resistance of the shunt must be high as compared with that of the external circuit, in order that only a small proportion of the total energy may be absorbed in the field.
Since the whole of the armature current passes round the field-magnet of the series machine, any alteration in the resistance of the external circuit will affect the excitation and also the voltage. A curve connecting together corresponding values of external current and terminal voltage for a given speed of rotation is known as the _external-characteristic_ of the machine; in its main features it has the same appearance as a curve of magnetic flux, but when the current exceeds a certain amount it begins to bend downwards and the voltage decreases. The reason for this will be found in the armature reaction at large loads, which gradually produces a more and more powerful demagnetizing effect, as the brushes are shifted forwards to avoid sparking; eventually the back ampere-turns overpower any addition to the field that would otherwise be due to the increased current flowing round the magnet. The "external characteristic" for a shunt machine has an entirely different shape. The field-magnet circuit being connected in parallel with the external circuit, the exciting current, if the applied voltage remains the same, is in no way affected by alterations in the resistance of the latter. As, however, an increase in the external current causes a greater loss of volts in the armature and a greater armature reaction, the terminal voltage, which is also the exciting voltage, is highest at no load and then diminishes. The fall is at first gradual, but after a certain critical value of the armature current is reached, the machine is rapidly demagnetized and loses its voltage entirely.
The last method of excitation, namely, _compound-winding_ (fig. 35), is a combination of the two preceding, and was first used by S.A. Varley and by C.F. Brush. If a machine is in the first instance shunt-wound, and a certain number of series-turns are added, the latter, since they carry the external current, can be made to counteract the effect which the increased external current would have in lowering the voltage of the simple shunt machine. The ampere-turns of the series winding must be such that they not only balance the increase of the demagnetizing back ampere-turns on the armature, but further increase the useful flux, and compensate for the loss of volts over their own resistance and that of the armature. The machine will then give for a constant speed a nearly constant voltage at its terminals, and the curve of the external characteristic becomes a straight line for all loads within its capacity. Since with most prime movers an increase of the load is accompanied by a drop in speed, this effect may also be counteracted; while, lastly, if the series-turns are still further increased, the voltage may be made to rise with an increasing load, and the machine is "over-compounded."
Commutation and sparking at the brushes.
At the initial moment when an armature coil is first short-circuited by the passage of the two sectors forming its ends under the contact surface of a brush, a certain amount of electromagnetic energy is stored up in its magnetic field as linked with the ampere-turns of the coil when carrying its full share of the total armature current. During the period of short-circuit this quantity of energy has to be dissipated as the current falls to zero, and has again to be re-stored as the current is reversed and raised to the same value, but in the opposite direction. The period of short-circuit as fixed by the widths of the brush and of the mica insulation between the sectors, and by the peripheral speed of the commutator is extremely brief, and only lasts on an average from (1/200)th to (1/1000)th of a second. The problem of sparkless commutation is therefore primarily a question of our ability to dissipate and to re-store the required amount of energy with sufficient rapidity.
An important aid towards the solution of this problem is found in the effect of the varying contact-resistance between the brush and the surfaces of the leading and trailing sectors which it covers. As the commutator moves under the brush, the area of contact which the brush makes with the leading sector diminishes, and the resistance between the two rises; conversely, the area of contact between the brush and the trailing sector increases and the resistance falls. This action tends automatically to bring the current through each sector into strict proportionality to the amount of its surface which is covered by the brush, and so to keep the current-density and the loss of volts over the contacts uniform and constant. As soon as the current-density in the two portions of the brush becomes unequal, a greater amount of heat is developed at the commutator surface, and this in the first place affords an additional outlet for the dissipation of the stored energy of the coil, while after reversal of the current it is the accompaniment of a re-storage of the required energy. This energy, as well as that which is spent in heating the coil, can in fact, in default of other sources, be derived through the action of the unequal current-density from the electrical output of the rest of the armature winding, and so only indirectly from the prime mover.
In practice, when the normal contact-resistance of the brushes is low relatively to the resistance of the coil, as is the case with metal brushes of copper or brass gauze, but little benefit can be obtained from the action of the varying contact-resistance. It exerts no appreciable effect until close towards the end of the period of short-circuit, and then only with such a high-current-density at the trailing edge of the leaving sector that at the moment of parting the brush-tip is fused, or its metal volatilized, and sparking has in fact set in. With such brushes, then, it becomes necessary to call in the aid of a reversing E.M.F. impressed upon the coil by the magnetic field through which it is moving. If such a reversing field comes into action while the current is still unreversed, its E.M.F. is opposed to the direction of the current, and the coil is therefore driving the armature forward as in a motor; it thus affords a ready means of rapidly dissipating part of the initial energy in the form of mechanical work instead of as heat. After the current has been reversed, the converse process sets in, and the prime mover directly expends mechanical energy not only in heating the coil, but also in storing up electromagnetic energy with a rapidity dependent upon the strength of the reversing field. The required direction of external field can be obtained in the dynamo by shifting the brushes forward, so that the short-circuited coil enters into the fringe of lines issuing from the leading pole-tip, i.e. by giving the brushes an "angle of lead." An objection to this process is that the main flux is thereby weakened owing to the belt of back ampere-turns which arises (_v. supra_). A still greater objection is that the amount of the angle of lead must be suited to the value of the load, the corrective power of copper brushes being very small if the reversing E.M.F. is not closely adjusted in proportion to the armature current.
On this account metal brushes have been almost entirely superseded by carbon moulded into hard blocks. With these, owing to their higher specific contact-resistance, a very considerable reversing effect can be obtained through the action of unequal current-density, and indeed in favourable cases complete sparklessness can be obtained throughout the entire range of load of the machine with a fixed position of the brushes. Yet if the work which they are called upon to perform exceeds certain limits, they tend to become overheated with consequent glowing or sparking at their tips, so that, wherever possible, it is advisable to reinforce their action by a certain amount of reversing field, the brushes being set so that its strength is roughly correct for, say, half load.
In the case of dynamos driven by steam-turbines, sparkless commutation is especially difficult to obtain owing to the high speed of rotation and the very short space of time in which the current has to be reversed. Special "reversing poles" then become necessary; these are wound with magnetizing coils in series with the main armature current, so that the strength of field which they yield is roughly proportional to the current which has to be reversed. These again may be combined with a "compensating winding" embedded in the pole-faces and carrying current in the opposite direction to the armature ampere-turns, so as to neutralize the cross effect of the latter and prevent distortion of the resultant field.
Heating effects.
From the moment that a dynamo begins to run with excited field, heat is continuously generated by the passage of the current through the windings of the field-magnet coils and the armature, as well as by the action of hysteresis and eddy currents in the armature and pole-pieces. Whether the source of the heat be in the field-magnet or in the armature, the mass in which it originates will continue to rise in temperature until such a difference of temperature is established between itself and the surrounding air that the rate at which the heat is carried off by radiation, convection and conduction is equal to the rate at which it is being generated. Evidently, then, the temperature which any part of the machine attains after a prolonged run must depend on the extent and effectiveness of the cooling surface from which radiation takes place, upon the presence or absence of any currents of air set up by the rotation of itself or surrounding parts, and upon the presence of neighbouring masses of metal to carry away the heat by conduction. In the field-magnet coils the rate at which heat is being generated is easily determined, since it is equal to the square of the current passing through them multiplied by their resistance. Further, the magnet is usually stationary, and only indirectly affected by draughts of air due to the rotating armature. Hence for machines of a given type and of similar proportions, it is not difficult to decide upon some method of reckoning the cooling surface of the magnet coils S_c, such that the rise of temperature above that of the surrounding air may be predicted from an equation of the form t deg. = kW/S_c, where W = the rate in watts at which heat is generated in the coils, and k is some constant depending upon the exact method of reckoning their cooling surface. As a general rule the cooling surface of a field-coil is reckoned as equal to the exposed outer surface of its wire, the influence of the end flanges being neglected, or only taken into account in the case of very short bobbins wound with a considerable depth of wire. In the case of the rotating armature a similar formula must be constructed, but with the addition of a factor to allow for the increase in the effectiveness of any given cooling surface due to the rotation causing convection currents in the surrounding air. Only experiment can determine the exact effect of this, and even with a given type of armature it is dependent on the number of poles, each of which helps to break up the air-currents, and so to dissipate the heat. For example, in two-pole machines with drum bar-armatures, if the cooling surface be reckoned as equal to the cylindrical exterior plus the area of the two ends, the heating coefficient for a peripheral speed of 1500 ft. per minute is less than half of that for the same armature when at rest. A further difficulty still meets the designer in the correct predetermination of the total loss of watts in an armature before the machine has been tested. It is made up of three separate items, namely, the copper loss in the armature winding, the loss by hysteresis in the iron, and the loss by eddy currents, which again may be divided into those in the armature bars and end-connexions, and those in the core and its end-plates. The two latter items are both dependent upon the speed of the machine; but whereas the hysteresis loss is proportional to the speed for a given density of flux in the armature, the eddy current loss is proportional to the square of the speed, and owing to this difference, the one loss can be separated from the other by testing an armature at varying speeds. Thus for a given rise of temperature, the question of the amount of current which can be taken out of an armature at different speeds depends upon the proportion which the hysteresis and eddy watts bear to the copper loss, and the ratio in which the effectiveness of the cooling surface is altered by the alteration in speed. Experimental data, again, can alone decide upon the amount of eddy currents that may be expected in given armatures, and caution is required in applying the results of one machine to another in which any of the conditions, such as the number of poles, density in the teeth, proportions of slot depth to width, &c., are radically altered.
It remains to add, that the rise of temperature which may be permitted in any part of a dynamo after a prolonged run is very generally placed at about 70 deg. Fahr. above the surrounding air. Such a limit in ordinary conditions of working leads to a final temperature of about 170 deg. Fahr., beyond which the durability of the insulation of the wires is liable to be injuriously affected. Upon some such basis the output of a dynamo in continuous working is rated, although for short periods of, say, two hours the normal full-load current of a large machine may be exceeded by some 25% without unduly heating the armature.
Uses of continuous current dynamos.
For the electro-deposition of metals or the electrolytic treatment of ores a continuous current is a necessity; but, apart from such use, the purposes from which the continuous-current dynamo is well adapted are so numerous that they cover nearly the whole field of electrical engineering, with one important exception. To meet these various uses, the pressures for which the machine is designed are of equally wide range; for the transmission of power over long distances they may be as high as 3000 volts, and for electrolytic work as low as five. Each electrolytic bath, with its leads, requires on an average only some four or five volts, so that even when several are worked in series the voltage of the dynamo seldom exceeds 60. On the other hand, the current is large and may amount to as much as from 1000 to 14,000 amperes, necessitating the use of two commutators, one at either end of the armature, in order to collect the current without excessive heating of the sectors and brushes. The field-magnets are invariably shunt-wound, in order to avoid reversal of the current through polarization at the electrodes of the bath. For incandescent lighting by glow lamps, the requirements of small isolated installations and of central stations for the distribution of electrical energy over large areas must be distinguished. For the lighting of a private house or small factory, the dynamo giving from 5 to 100 kilo-watts of output is commonly wound for a voltage of 100, and is driven by pulley and belt from a gas, oil or steam-engine; or, if approaching the higher limit above mentioned, it is often directly coupled to the crank-shaft of the steam-engine. If used in conjunction with an accumulator of secondary cells, it is shunt-wound, and must give the higher voltage necessary to charge the battery; otherwise it is compound-wound, in order to maintain the pressure on the lamps constant under all loads within its capacity. The compound-wound dynamo is likewise the most usual for the lighting of steamships, and is then directly coupled to its steam-engine; its output seldom exceeds 100 kilo-watts, at a voltage of 100 or 110. For larger installations a voltage of 250 is commonly used, while for central-station work, economy in the distributing mains dictates a higher voltage, especially in connexion with a three-wire system; the larger dynamos may then give 500 volts, and be connected directly across the two outer wires. A pair of smaller machines coupled together, and each capable of giving 250 volts, are often placed in series across the system, with their common junction connected to the middle wire; the one which at any time is on the side carrying the smaller current will act as a motor and drive the other as a dynamo, so as to balance the system. The directly-coupled steam dynamo may be said to have practically displaced the belt- or rope-driven sets which were formerly common in central stations. The generating units of the central station are arranged in progressive sizes, rising from, it may be, 250 or 500 horse-power up to 750 or 1000, or in large towns to as much as 5000 horse-power. If for lighting only, they are usually shunt-wound, the regulation of the voltage, to keep the pressure constant on the distributing system under the gradual changes of load, being effected by variable resistances in the shunt circuit of the field-magnets.
Generators used for supplying current to electric tramways are commonly wound for 500 volts at no load and are over-compounded, so that the voltage rises to 550 volts at the maximum load, and thus compensates for the loss of volts over the transmitting lines. For arc lighting it was formerly usual to employ a class of dynamo which, from the nature of its construction, was called an "open-coil" machine, and which gave a unidirectional but pulsating current. Of such machines the Brush and Thomson-Houston types were very widely used; their E.M.F. ranged from 2000 to 3000 volts for working a large number of arcs in series, and by means of special regulators their current was maintained constant over a wide range of voltage. But as their efficiency was low and they could not be applied to any other purpose, they have been largely superseded in central stations by closed-coil dynamos or alternators, which can also be used for incandescent lighting. In cases where the central station is situated at some distance from the district to which the electric energy is to be supplied, voltages from 1000 to 2000 are employed, and these are transformed down at certain distributing centres by continuous-current transformers (see TRANSFORMERS and ELECTRICITY SUPPLY). These latter machines are in reality motor-driven dynamos, and hence are also called _motor-generators_; the armatures of the motor and dynamo are often wound on the same core, with a commutator at either end, the one to receive the high-pressure motor current, and the other to collect the low-pressure current furnished by the dynamo.
In all large central stations it is necessary that the dynamos should be capable of being run _in parallel_, so that their outputs may be combined on the same "omnibus bars" and thence distributed to the network of feeders. With simple shunt-wound machines this is easily effected by coupling together terminals of like sign when the voltage of the two or more machines are closely equal. With compound-wound dynamos not only must the external terminals of like sign be coupled together, but the junctions of the brush leads with the series winding must be connected by an "equalizing" lead of low resistance; otherwise, should the E.M.F. of one machine for any reason fall below the voltage of the omnibus bars, there is a danger of its polarity being reversed by a back current from the others with which it is in parallel.
Owing to the necessary presence in the continuous-current dynamo of the commutator, with its attendant liability to sparking at the brushes, and further, owing to the difficulty of insulating the rotating armature wires, a pressure of 3000 volts has seldom been exceeded in any one continuous-current machine, and has been given above as the limiting voltage of the class. If therefore it is required to work with higher pressures in order to secure economy in the transmitting lines, two or more machines must be coupled _in series_ by connecting together terminals which are of unlike sign.[19] The stress of the total voltage may still fall on the insulation of the winding from the body of the machine; hence for high-voltage transmission of power over very long distances, the continuous-current dynamo in certain points yields in convenience to the alternator. In this there is no commutator, the armature coils may be stationary and can be more thoroughly insulated, while further, if it be thought undesirable to design the machine for the full transmitting voltage, it is easy to wind the armature for a low pressure; this can be subsequently transformed up to a high pressure by means of the alternating-current transformer, which has stationary windings and so high an efficiency that but little loss arises from its use. With these remarks, the transition may be made to the fuller discussion of the alternator.
_Alternators._
Frequency.
The frequency employed in alternating-current systems for distributing power and light varies between such wide limits as 25 and 133; yet in recent times the tendency has been towards standard frequencies of 25, 50 and 100 as a maximum. High frequencies involve more copper in the magnet coils, owing to the greater number of poles, and a greater loss of power in their excitation, but the alternator as a whole is somewhat lighter, and the transformers are cheaper. On the other hand, high frequency may cause prejudicial effects, due to the inductance and capacity of the distributing lines; and in asynchronous motors used on polyphase systems the increased number of poles necessary to obtain reasonable speeds reduces their efficiency, and is otherwise disadvantageous, especially for small horse-powers. A frequency lower than 40 is, however, not permissible where arc lighting is to form any considerable portion of the work and is to be effected by the alternating current without rectification, since below this value the eye can detect the periodic alteration in the light as the carbons alternately cool and become heated. Thus for combined lighting and power 50 or 60 are the most usual frequencies; but if the system is designed solely or chiefly for the distribution of power, a still lower frequency is preferable. On this account 25 was selected by the engineers for the Niagara Falls power transmission, after careful consideration of the problem, and this frequency has since been widely adopted in similar cases.
Alternator construction.
The most usual type of heteropolar alternator has an internal rotating field-magnet system, and an external stationary armature, as in fig. 10. The coils of the armature, which must for high voltages be heavily insulated, are then not subjected to the additional stresses due to centrifugal force; and further, the collecting rings which must be attached to the rotating portion need only transmit the exciting current at a low voltage.
The homopolar machine possesses the advantages that only a single exciting coil is required, whatever the number of polar projections, and that both the armature and field-magnet coils may be stationary. From fig. 8 it will be seen that it is not essential that the exciting coil should revolve with the internal magnet, but it may be supported from the external stationary armature while still embracing the central part of the rotor. The E.M.F. is set up in the armature coils through the periodic variation of the flux through them as the iron projections sweep past, and these latter may be likened to a number of "keepers," which complete the magnetic circuit. From the action of the rotating iron masses they may also be considered as the inducing elements or "inductors," and the homopolar machine is thence also known as the "inductor alternator." If the end of the rotor marked S in fig. 8 is split up into a number of S polar projections similar to the N poles, a second set of armature coils may be arranged opposite to them, and we obtain an inductor alternator with double armature. Or the polar projections at the two ends may be staggered, and a single armature winding be passed straight through the armature, as in fig. 36, which shows at the side the appearance of the revolving inductor with its crown of polar projections in one ring opposite to the gaps between the polar projections of the other ring. But in spite of its advantage of the single stationary exciting coil, the inductor alternator has such a high degree of leakage, and the effect of armature reaction is so detrimental in it, that the type has been gradually abandoned, and a return has been almost universally made to the heteropolar alternator with internal poles radiating outwards from a circular yoke-ring. The construction of a typical machine of this class is illustrated in fig. 37.
Since the field-magnet coils rotate, they must be carefully designed to withstand centrifugal force, and are best composed of flat copper strip wound on edge with thin insulation between adjacent layers. The coil is secured by the edges of the pole-shoes which overhang the pole and tightly compress the coil against the yoke-ring; the only effect from centrifugal force is then to compress still further the flat turns of copper against the pole-shoes without deformation. The poles are either of cast steel of circular or oblong section, bolted to the rim of the yoke-ring, or are built up of thin laminations of sheet steel. When the peripheral speed is very high, the yoke-ring will be of cast steel or may itself be built up of sheet steel laminations, this material being reliable and easily tested to ensure its sound mechanical strength. If the armature slots are open, the pole-pieces will in any case be laminated to reduce the eddy currents set up by the variation of the flux-density.
Owing to the great number of poles[20] of the alternator when driven by a reciprocating steam-engine, the diameter of its rotor is usually larger and its length less than in the continuous-current dynamo of corresponding output. The support of the armature core when of large diameter is therefore a more difficult problem, since, apart from any magnetic strains to which it may be subjected, its own weight tends to deform it. The segmental core-disks are usually secured to the internal circumference of a circular cast iron frame; the latter has a box section of considerable radial depth to give stiffness to it, and the disks are tightly clamped between internal flanges, one being a fixed part of the frame and the other loose, with transverse bolts passing right through from side to side (fig. 37). In order to lessen the weight of the structure and its expense in material, the cast iron frame has in some cases been entirely dispensed with, and braced tie-rods have been used to render the effective iron of the armature core-disks self-supporting.
Owing to the high speed of the turbo-alternator, its rotor calls for the utmost care in its design to withstand the effect of centrifugal force without any shifting of the exciting coils, and to secure a perfect balance.
The appearance of the armature of a typical three-phase alternator is illustrated in fig. 38, which shows a portion of the lower half after removal of the field-magnet.
With open slots the coils, after being wound on formers to the required shape, are thoroughly impregnated with insulating compound, dried, and after a further wrapping with several layers of insulating material, finally pressed into the slots together with a sheet of leatheroid or flexible micanite. The end-connexions of each group of coils of one phase project straight out from the slots or are bent upwards alternately with those of the other phases, so that they may clear one another (fig. 37). A wooden wedge driven into a groove at the top of each slot is often used to lock the coil in place. With slots nearly closed at the top, the coils are formed by hand by threading the wire through tubes of micanite or specially prepared paper lining the slots; or with single-turn loops, stout bars of copper of [U]-shape can be driven through the slots and closed by soldered connexions at the other end.
Shape of E.M.F. curve.
The first experimental determination of the shape of the E.M.F. curve of an alternator was made by J. Joubert in 1880. A revolving contact-maker charged a condenser with the E.M.F. produced by the armature at a particular instant during each period. The condenser was discharged through a ballistic galvanometer, and from the measured throw the instantaneous E.M.F. could be deduced. The contact-maker was then shifted through a small angle, and the instantaneous E.M.F. at the new position corresponding to a different moment in the period was measured; this process was repeated until the E.M.F. curve for a complete period could be traced. Various modifications of the same principle have since been used, and a form of "oscillograph" (q.v.) has been perfected which is well adapted for the purpose of tracing the curves both of E.M.F. and of current. The machine on which Joubert carried out his experiments was a Siemens disk alternator having no iron in its armature, and it was found that the curve of E.M.F. was practically identical with a sine curve. The same law has also been found to hold true for a smooth-core ring or drum armature, but the presence of the iron core enables the armature current to produce greater distorting effect, so that the curves under load may vary considerably from their shape at no load. In toothed armatures, the broken surface of the core, and the still greater reaction from the armature current, may produce wide variations from the sine law, the general tendency being to give the E.M.F. curve a more peaked form. The great convenience of the assumption that the E.M.F. obeys the sine law has led to its being very commonly used as the basis for the mathematical analysis of alternator problems; but any deductions made from this premiss require to be applied with caution if they are likely to be modified by a different shape of the curve. Further, the same alternator will give widely different curves even of E.M.F., and still more so of current, according to the nature of the external circuit to which it is connected. As will be explained later, the phase of the current relatively to the E.M.F. depends not only on the inductance of the alternator itself, but also upon the inductance and capacity of the external circuit, so that the same current will produce different effects according to the amount by which it lags or leads. The question as to the relative advantages of differently shaped E.M.F. curves has led to much discussion, but can only be answered by reference to the nature of the work that the alternator has to do--i.e. whether it be arc lighting, motor driving, or incandescent lighting through transformers. The shape of the E.M.F. curve is, however, of great importance in one respect, since upon it depends the ratio of the maximum instantaneous E.M.F. to the effective value, and the insulation of the entire circuit, both external and internal, must be capable of withstanding the maximum E.M.F. While the maximum value of the sine curve is [root]2 or 1.414 times the effective value, the maximum value of a [Lambda] curve is 1.732 times the effective value, so that for the same effective E.M.F. the armature wires must not only be more heavily insulated than in the continuous-current dynamo, but also the more peaked the curve the better must be the insulation.
Excitation.
Since an alternating current cannot be used for exciting the field-magnet, recourse must be had to some source of a direct current. This is usually obtained from a small auxiliary continuous-current dynamo, called an _exciter_, which may be an entirely separate machine, separately driven and used for exciting several alternators, or may be driven from the alternator itself; in the latter case the armature of the exciter is often coupled directly to the rotating shaft of the alternator, while its field-magnet is attached to the bed-plate. Although separate excitation is the more usual method, the alternator can also be made self-exciting if a part or the whole of the alternating current is "rectified," and thus converted into a direct current.
Quarter-phase alternators.
The general idea of the polyphase alternator giving two or more E.M.F.'s of the same frequency, but displaced in phase, has been already described. The several phases may be entirely independent, and such was the case with the early polyphase machines of Gramme, who used four independent circuits, and also in the large two-phase alternators designed by J.E.H. Gordon in 1883. If the phases are thus entirely separate, each requires two collector rings and two wires to its external circuit, i.e. four in all for two-phase and six for three-phase machines. The only advantage of the polyphase machine as thus used is that the whole of the surface of the armature core may be efficiently covered with winding, and the output of the alternator for a given size be thereby increased. It is, however, also possible so to interlink the several circuits of the armature that the necessary number of transmitting lines to the external circuits may be reduced, and also the weight of copper in them for a given loss in the transmission.[21] The condition which obviously must be fulfilled, for such interlinking of the phases to be possible, is that in the lines which are to meet at any common junction the algebraic sum of the instantaneous currents, reckoned as positive if away from such junction and as negative if towards it, must be zero. Thus if the phases be diagrammatically represented by the relative angular position of the coils in fig. 39, the current in the coils A and B differs in phase from the current in the coils C and D by a quarter of a period or 90 deg.; hence if the two wires b and d be replaced by the single wire bd, this third wire will serve as a common path for the currents of the two phases either outwards or on their return. At any instant the value of the current in the third wire must be the vector sum of the two currents in the other wires, and if the shape of the curves of instantaneous E.M.F. and current are identical, and are assumed to be sinusoidal, the effective value of the current in the third wire will be the vector sum of the effective values of the currents in the other wires; in other words, if the system is balanced, the effective current in the third wire is [root]2, or 1.414 times the current in either of the two outer wires. Since the currents of the two phases do not reach their maximum values at the same time, the sectional area of the third wire need not be twice that of the others; in order to secure maximum efficiency by employing the same current density in all three wires, it need only be 40% greater than that of either of the outer wires. The effective voltage between the external leads may in the same way be calculated by a vector diagram, and with the above _star connexion_ the voltage between the outer pair of wires a and c is [root]2, or 1.414 times the voltage between either of the outer wires and the common wire bd. Next, if the four coils are joined up into a continuous helix, just as in the winding of a continuous-current machine, four wires may be attached to equidistant points at the opposite ends of two diameters at right angles to each other (fig. 40). Such a method is known as the _mesh connexion_, and gives a perfectly symmetrical four-phase system of distribution. Four collecting rings are necessary if the armature rotates, and there is no saving in copper in the transmitting lines; but the importance of the arrangement lies in its use in connexion with rotary converters, in which it is necessary that the winding of the armature should form a closed circuit. If e = the effective voltage of one phase A, the voltage between any pair of adjacent lines in the diagram is e, and between m and o or n and p is e [root]2. The current in any line is the resultant of the currents in the two phases connected to it, and its effective value is c [root]2, where c is the current of one phase.
Three-phase alternators.
When we pass to machines giving three phases differing by 120 deg., the same methods of star and mesh connexion find their analogies. If the current in coil A (fig. 41) is flowing away from the centre, and has its maximum value, the currents in coils B and C are flowing towards the centre, and are each of half the magnitude of the current in A; the algebraic sum of the currents is therefore zero, and this will also be the case for all other instants. Hence the three coils can be united together at the centre, and three external wires are alone required. In this star or "Y" connexion, if e be the effective voltage of each phase, or the voltage between any one of the three collecting rings and the common connexion, the volts between any pair of transmitting lines will be E = e [root]3 (fig. 41); if the load be balanced, the effective current C in each of the three lines will be equal, and the total output in watts will be W = 3Ce = 3CE/[root]3 = 1.732 EC, or 1.732 times the product of the effective voltage between the lines and the current in any single line. Next, if the three coils are closed upon themselves in a mesh or _delta_ fashion (fig. 42), the three transmitting wires may be connected to the junctions of the coils (by means of collecting rings if the armature rotates). The voltage E between any pair of wires is evidently that generated by one phase, and the current in a line wire is the resultant of that in two adjacent phases; or in a balanced system, if c be the current in each phase, the current in the line wire beyond a collecting ring is C = c [root]3, hence the watts are W = 3cE = 3CE/[root]3 = 1.732 EC, as before. Thus any three-phase winding may be changed over from the star to the delta connexion, and will then give 1.732 times as much current, but only 1/1.732 times the voltage, so that the output remains the same.
Armature reaction in alternators
The "armature reaction" of the alternator, when the term is used in its widest sense to cover all the effects of the alternating current in the armature as linked with a magnetic circuit or circuits, may be divided into three items which are different in their origin and consequences. In the first place the armature current produces a self-induced flux in local circuits independent of the main magnetic circuit, as e.g. linked with the ends of the coils as they project outwards from the armature core; such lines may be called "secondary leakage," of which the characteristic feature is that its amount is independent of the position of the coils relatively to the poles. The alternations of this flux give rise to an inductive voltage lagging 90 deg. behind the phase of the current, and this leakage or reactance voltage must be directly counterbalanced electrically by an equal component in the opposite sense in the voltage from the main field. The second and third elements are more immediately magnetic and are entirely dependent upon the position of the coils in relation to the poles and in relation to the phase of the current which they then carry. When the side of a drum coil is immediately under the centre of a pole, its ampere-turns are cross-magnetizing, i.e. produce a distortion of the main flux, displacing its maximum density to one or other edge of the pole. When the coil-side is midway between the poles and the axes of coil and pole coincide, the coil stands exactly opposite to the pole and embraces the same magnetic circuit as the field-magnet coils; its turns are therefore directly magnetizing, either weakening or strengthening the main flux according to the direction of the current. In intermediate positions the ampere-turns of the coil gradually pass from cross to direct and vice versa. When the instantaneous values of either the cross or direct magnetizing effect are integrated over a period and averaged, due account being taken of the number of slots per coil-side and of the different phases of the currents in the polyphase machine, expressions are obtained for the equivalent cross and direct ampere-turns of the armature as acting upon a pair of poles. For a given winding and current, the determining factor in either the one or the other is found to be the relative phase angle between the axis of a coil in its position when carrying the maximum current and the centre of a pole, the transverse reaction being proportional to the cosine of this angle, and the direct reaction to its sine. If the external circuit is inductive, the maximum value of the current lags behind the E.M.F. and so behind the centre of the pole; such a negative angle of lag causes the direct magnetizing turns to become back turns, directly weakening the main field and lowering the terminal voltage. Thus, just as in the continuous-current dynamo, for a given voltage under load the excitation between the pole-pieces X_p must not only supply the net excitation required over the air-gaps, armature core and teeth, but must also balance the back ampere-turns X_b of the armature.
Evidently therefore the characteristic curve connecting armature current and terminal volts will with a constant exciting current depend on the nature of the load, whether inductive or non-inductive, and upon the amount of inductance already possessed by the armature itself. With an inductive load it will fall more rapidly from its initial maximum value, or, conversely, if the initial voltage is to be maintained under an increasing load, the exciting current will have to be increased more than if the load were non-inductive. In practical working many disadvantages result from a rapid drop of the terminal E.M.F. under increasing load, so that between no load and full load the variation in terminal voltage with constant excitation should not exceed 15%. Thus the output of an alternator is limited either by its heating or by its armature reaction, just as is the output of a continuous-current dynamo; in the case of the alternator, however, the limit set by armature reaction is not due to any sparking at the brushes, but to the drop in terminal voltage as the current is increased, and the consequent difficulty in maintaining a constant potential on the external circuit.
The coupling of alternators.
The joint operation of several alternators so that their outputs may be delivered into the same external circuit is sharply distinguished from the corresponding problem in continuous-current dynamos by the necessary condition that they must be in synchronism, i.e. not only must they be so driven that their frequency is the same, but their E.M.F.'s must be in phase or, as it is also expressed, the machines must be in step. Although in practice it is impossible to run two alternators in series unless they are rigidly coupled together--which virtually reduces them to one machine--two or more machines can be run in parallel, as was first described by H. Wilde in 1868 and subsequently redemonstrated by J. Hopkinson and W.G. Adams in 1884. Their E.M.F.'s should be as nearly as possible in synchronism, but, as contrasted with series connexion, parallel coupling gives them a certain power of recovery if they fall out of step, or are not in exact synchronism when thrown into parallel. In such circumstances a synchronizing current passes between the two machines, due to the difference in their instantaneous pressures; and as this current agrees in phase more nearly with the leading than with the lagging machine, the former machine does work as a generator on the latter as a motor. Hence the lagging machine is accelerated and the leading machine is retarded, until their frequencies and phase are again the same.
Uses of alternators.
The chief use of the alternator has already been alluded to. Since it can be employed to produce very high pressures either directly or through the medium of transformers, it is specially adapted to the electrical transmission of energy over long distances.[22] In the early days of electric lighting, the alternate-current system was adopted for a great number of central stations; the machines, designed to give a pressure of 2000 volts, supplied transformers which were situated at considerable distances and spread over large areas, without an undue amount of copper in the transmitting lines. While there was later a tendency to return to the continuous current for central stations, owing to the introduction of better means for economizing the weight of copper in the mains, the alternating current again came into favour, as rendering it possible to place the central station in some convenient site far away from the district which it was to serve. The pioneer central station in this direction was the Deptford station of the London Electric Supply Corporation, which furnished current to the heart of London from a distance of 7 m. In this case, however, the alternators were single-phase and gave the high pressure of 10,000 volts immediately, while more recently the tendency has been to employ step-up transformers and a polyphase system. The advantage of the latter is that the current, after reaching the distant sub-stations, can be dealt with by rotary converters, through which it is transformed into a continuous current. The alternator is also used for welding, smelting in electric furnaces, and other metallurgical processes where heating effects are alone required; the large currents needed therein can be produced without the disadvantage of the commutator, and, if necessary, transformers can be interposed to lower the voltage and still further increase the current. The alternating system can thus meet very various needs, and its great recommendation may be said to lie in the flexibility with which it can supply electrical energy through transformers at any potential, or through rotary converters in continuous-current form.
AUTHORITIES.--For the further study of the dynamo, the following may be consulted, in addition to the references already given:--
_General_: S.P. Thompson, _Dynamo-Electric Machinery--Continuous-Current Machines_ (1904), _Alternating-Current Machinery_ (1905, London); G. Kapp, _Dynamos, Alternators and Transformers_ (London, 1893); _Id., Electric Transmission of Energy_ (London, 1894); Id., _Dynamo Construction; Electrical and Mechanical_ (London, 1899); H.F. Parshall and H.M. Hobart, _Electric Generators_ (London, 1900); C.C. Hawkins and F. Wallis, _The Dynamo_ (London, 1903); E. Arnold, _Konstruktionstafeln fuer den Dynamobau_ (Stuttgart, 1902); C.P. Steinmetz, _Elements of Electrical Engineering_ (New York, 1901).
_Continuous-Current Dynamos_: J. Fischer-Hinnen, _Continuous-Current Dynamos_ (London, 1899); E. Arnold, _Die Gleichstrommaschine_ (Berlin, 1902); F. Niethammer, _Berechnung und Konstruktion der Gleichstrommaschinen und Gleichstrommotoren_ (Stuttgart, 1904).
_Alternators_: D.C. Jackson and J.P. Jackson, _Alternating Currents and Alternating Current Machinery_ (New York, 1903); J.A. Fleming, _The Alternate Current Transformer_ (London, 1899); C.P. Steinmetz, _Alternating Current Phenomena_ (New York, 1900); E. Arnold, _Die Wechselstromtechnik_ (Berlin, 1904); S.P. Thompson, _Polyphase Electric Currents_ (London, 1900); A. Stewart, _Modern Polyphase Machinery_ (London, 1906); M. Oudin, _Standard Polyphase Apparatus and Systems_ (New York, 1904). (C. C. H.)
FOOTNOTES:
[1] _Experimental Researches in Electricity_, series ii. Sec. 6, pars. 256, 259-260, and series xxviii. Sec. 34.
[2] _Ibid._ series i. Sec. 4, pars. 84-90.
[3] "On the Physical Lines of Magnetic Force," _Phil. Mag._, June 1852.
[4] Faraday, _Exp. Res._ series xxviii. Sec. 34, pars. 3104, 3114-3115.
[5] _Id._, ib. series i. Sec. 4, pars. 114-119.
[6] _Id._, ib. series ii. Sec. 6, pars. 211, 213; series xxviii. Sec. 34, par. 3152.
[7] Invented by Nikola Tesla (_Elec. Eng._ vol. xiii. p. 83. Cf. Brit. Pat. Spec. Nos. 2801 and 2812, 1894). Several early inventors, e.g. Salvatore dal Negro in 1832 (_Phil. Mag._ third series, vol. i. p. 45), adopted reciprocating or oscillatory motion, and this was again tried by Edison in 1878.
[8] The advantage to be obtained by making the poles closely embrace the armature core was first realized by Dr Werner von Siemens in his "shuttle-wound" armature (Brit. Pat. No. 2107, 1856).
[9] _Nuovo Cimento_ (1865), 19, 378.
[10] Brit. Pat. No. 1668 (1870); _Comptes rendus_ (1871), 73, 175.
[11] _Ann. Chim. Phys._ l. 322.
[12] Ibid. li. 76. Since in H. Pixii's machine the armature was stationary, while both magnet and commutator rotated, four brushes were used, and the arrangement was not so simple as the split-ring described above, although the result was the same. J. Saxton's machine (1833) and E.M. Clarke's machine (1835, see Sturgeon's _Annals of Electricity_, i. 145) were similar to one another in that a unidirected current was obtained by utilizing every alternate half-wave of E.M.F., but the former still employed mercury collecting cups, while the latter employed metal brushes. W. Sturgeon in 1835 followed Pixii in utilizing the entire wave of E.M.F., and abandoned the mercury cups in favour of metal brushes pressing on four semicircular disks (_Scientific Researches_, p. 252). The simple split-ring is described by Sir C. Wheatstone and Sir W.F. Cooke in their Patent No. 8345 (1840).
[13] By the "leading" side of the tooth or of an armature coil or sector is to be understood that side which first enters under a pole after passing through the interpolar gap, and the edge of the pole under which it enters is here termed the "leading" edge as opposed to the "trailing" edge or corner from under which a tooth or coil emerges into the gap between the poles; cf. fig. 30, where the leading and trailing pole-corners are marked ll and tt.
[14] Such was the arrangement of Wheatstone's machine (Brit. Pat. No. 9022) of 1841, which was the first to give a more nearly "continuous" current, the number of sections and split-rings being five.
[15] Its development from the split-ring was due to Pacinotti and Gramme (Brit. Pat. No. 1668, 1870) in connexion with their ring armatures.
[16] And extended by G. Kapp, "On Modern Continuous-Current Dynamo-Electric Machines," _Proc. Inst. C.E._ vol. lxxxiii. p. 136.
[17] Drs J. and E. Hopkinson, "Dynamo-Electric Machinery," Phil. Trans., May 6, 1886; this was further expanded in a second paper on "Dynamo-Electric Machinery," _Proc. Roy. Soc._, Feb. 15, 1892, and both are reprinted in _Original Papers on Dynamo-Machinery and Allied Subjects_.
[18] _Exp. Res._, series i. Sec. 4, par. 111. In 1845 Wheatstone and Cooke patented the use of "voltaic" magnets in place of permanent magnets (No. 10,655).
[19] Between Moutiers and Lyons, a distance of 115 m., energy is transmitted on the Thury direct-current system at a maximum pressure of 60,000 volts. Four groups of machines in series are employed, each group consisting of four machines in series; the rated output of each component machine is 75 amperes at 3900 volts or 400 h.p. A water turbine drives two pairs of such machines through an insulating coupling, and the sub-base of each pair of machines is separately insulated from earth, the foundation being also of special insulating materials.
[20] For experiments on high-frequency currents, Nikola Tesla constructed an alternator having 384 poles and giving a frequency of about 10,000 (_Journ. Inst. Elec. Eng._ 1892, 21, p. 82). The opposite extreme is found in alternators directly coupled to the Parsons steam-turbine, in which, with a speed of 3000 revs. per min., only two poles are required to give a frequency of 50. By a combination of a Parsons steam-turbine running at 12,000 revs. per min. with an alternator of 140 poles a frequency of 14,000 has been obtained (_Engineering_, 25th of August 1899). For description of an experimental machine for 10,000 cycles per second when running at 3000 revs. per min., see _Trans. Amer. Inst. Elect. Eng._ vol. xxiii. p. 417.
[21] As in the historical transmission of energy from Lauffen to Frankfort (1891).
[22] In the pioneer three-phase transmission between Laufen and Frankfort (_Electrician_, vol. xxvi. p. 637, and xxvii. p. 548), the three-phase current was transformed up from about 55 to 8500 volts, the distance being 110 m. A large number of installations driven by water power are now at work, in which energy is transmitted on the alternating-current system over distances of about 100 m. at pressures ranging from 20,000 to 67,000 volts.
DYNAMOMETER (Gr. [Greek: dynamis], strength, and [Greek: metron], a measure), an instrument for measuring force exerted by men, animals and machines. The name has been applied generally to all kinds of instruments used in the measurement of a force, as for example electric dynamometers, but the term specially denotes apparatus used in connexion with the measurement of work, or in the measurement of the horse-power of engines and motors. If P represent the average value of the component of a force in the direction of the displacement, s, of its point of application, the product Ps measures the work done during the displacement. When the force acts on a body free to turn about a fixed axis only, it is convenient to express the work done by the transformed product T[theta], where T is the average turning moment or torque acting to produce the displacement [theta] radians. The apparatus used to measure P or T is the dynamometer. The factors s or [theta] are observed independently. Apparatus is added to some dynamometers by means of which a curve showing the variations of P on a distance base is drawn automatically, the area of the diagram representing the work done; with others, integrating apparatus is combined, from which the work done during a given interval may be read off directly. It is convenient to distinguish between absorption and transmission dynamometers. In the first kind the work done is converted into heat; in the second it is transmitted, after measurement, for use.
_Absorption Dynamometers._--Baron Prony's dynamometer (_Ann. Chim. Phys._ 1821, vol. 19), which has been modified in various ways, consists in its original form of two symmetrically shaped timber beams clamped to the engine-shaft. When these are held from turning, their frictional resistance may be adjusted by means of nuts on the screwed bolts which hold them together until the shaft revolves at a given speed. To promote smoothness of action, the rubbing surfaces are lubricated. A weight is moved along the arm of one of the beams until it just keeps the brake steady midway between the stops which must be provided to hold it when the weight fails to do so. The general theory of this kind of brake is as follows:-Let F be the whole frictional resistance, r the common radius of the rubbing surfaces, W the force which holds the brake from turning and whose line of action is at a perpendicular distance R from the axis of the shaft, N the revolutions of the shaft per minute, [omega] its angular velocity in radians per second; then, assuming that the adjustments are made so that the engine runs steadily at a uniform speed, and that the brake is held still, clear of the stops and without oscillation, by W, the torque T exerted by the engine is equal to the frictional torque Fr acting at the brake surfaces, and this is measured by the statical moment of the weight W about the axis of revolution; that is--
T = Fr = WR. (1)
Hence WR measures the torque T.
If more than one force be applied to hold the brake from turning, Fr, and therefore T, are measured by the algebraical sum of their individual moments with respect to the axis. If the brake is not balanced, its moment about the axis must be included. Therefore, quite generally,
T = [Sigma]WR. (2)
The factor [theta] of the product T[theta] is found by means of a revolution counter. The power of a motor is measured by the rate at which it works, and this is expressed by T[omega] = T2[pi]N/60 in foot-pounds per second, or T2[pi]N/33,000 in horse-power units. The latter is commonly referred to as the "brake horse-power." The maintenance of the conditions of steadiness implied in equation (1) depends upon the constancy of F, and therefore of the coefficient of friction mu between the rubbing surfaces. The heating at the surfaces, the variations in their smoothness, and the variations of the lubrication make [mu] continuously variable, and necessitate frequent adjustment of W or of the nuts. J.V. Poncelet (1788-1867) invented a form of Prony brake which automatically adjusted its grip as [mu] changed, thereby maintaining F constant.
The principle of the compensating brake devised by J.G. Appold (1800-1865) is shown in fig. 1. A flexible steel band, lined with wood blocks, is gripped on the motor fly-wheel or pulley by a screw A, which, together with W, is adjusted to hold the brake steady. Compensation is effected by the lever L inserted at B. This has a slotted end, engaged by a pin P fixed to the framing, and it will be seen that its action is to slacken the band if the load tends to rise and to tighten it in the contrary case. The external forces holding the brake from turning are W, distant R from the axis, and the reaction, W1 say, of the lever against the fixed pin P, distant R1 from the axis. The moment of W1 may be positive or negative. The torque T at any instant of steady running is therefore {WR +- W1R1}.
Lord Kelvin patented a brake in 1858 (fig. 2) consisting of a rope or cord wrapped round the circumference of a rotating wheel, to one end of which is applied a regulated force, the other end being fixed to a spring balance. The ropes are spaced laterally by the blocks B, B, B, B, which also serve to prevent them from slipping sideways. When the wheel is turning in the direction indicated, the forces holding the band still are W, and p, the observed pull on the spring balance. Both these forces usually act at the same radius R, the distance from the axis to the centre line of the rope, in which case the torque T is (W - p)R, and consequently the brake horse-power is
(W - p)R x 2[pi]N -----------------. 33,000
When mu changes the weight W rises or falls against the action of the spring balance until a stable condition of running is obtained. The ratio {W/p} is given by e^{ mu[theta]}, where e = 2.718; mu is the coefficient of friction and [theta] the angle, measured in radians, subtended by the arc of contact between the rope and the wheel. In fig. 2 [theta] = 2[pi]. The ratio W/p increases very rapidly as [theta] is increased, and therefore, by making [theta] sufficiently large, p may conveniently be made a small fraction of W, thereby rendering errors of observation of the spring balance negligible. Thus this kind of brake, though cheap to make, is, when [theta] is large enough, an exceedingly accurate measuring instrument, readily applied and easily controlled. It has come into very general use in recent years, and has practically superseded the older forms of block brakes.
It is sometimes necessary to use water to keep the brake wheel cool. Engines specially designed for testing are usually provided with a brake wheel having a trough-shaped rim. Water trickles continuously into the trough, and the centrifugal action holds it as an inside lining against the rim, where it slowly evaporates.
Fig. 3 shows a band-brake invented by Professor James Thomson, suitable for testing motors exerting a constant torque (see _Engineering_, 22nd October 1880). To maintain e^{ mu[theta]} constant, compensation for variation of [mu] is made by inversely varying [theta]. A and B are fast and loose pulleys, and the brake band is placed partly over the one and partly over the other. Weights W and w are adjusted to the torque. The band turns with the fast pulley if [mu] increase, thereby slightly turning the loose pulley, otherwise at rest, until [theta] is adjusted to the new value of [mu]. This form of brake was also invented independently by J.A.M.L. Carpentier, and the principle has been used in the Raffard brake. A self-compensating brake of another kind, by Marcel Deprez, was described with Carpentier's in 1880 (_Bulletin de la societe d'encouragement_, Paris). W.E. Ayrton and J. Perry used a band or rope brake in which compensation is effected by the pulley drawing in or letting out a part of the band or rope which has been roughened or in which a knot has been tied.
In an effective water-brake invented by W. Froude (see _Proc. Inst. M. E._ 1877), two similar castings, A and B, each consisting of a boss and circumferential annular channel, are placed face to face on a shaft, to which B is keyed, A being free (fig. 4). A ring tube of elliptical section is thus formed. Each channel is divided into a series of pockets by equally spaced vanes inclined at 45 deg.. When A is held still, and B rotated, centrifugal action sets up vortex currents in the water in the pockets; thus a continuous circulation is caused between B and A, and the consequent changes of momentum give rise to oblique reactions. The moments of the components of these actions and reactions in a plane to which the axis of rotation is at right angles are the two aspects of the torque acting, and therefore the torque acting on B through the shaft is measured by the torque required to hold A still. Froude constructed a brake to take up 2000 H.P. at 90 revs. per min. by duplicating this apparatus. This replaced the propeller of the ship whose engines were to be tested, and the outer casing was held from turning by a suitable arrangement of levers carried to weighing apparatus conveniently disposed on the wharf. The torque corresponding to 2000 H.P. at 90 revs. per min. is 116,772 foot-pounds, and a brake 5 ft. in diameter gave this resistance. Thin metal sluices were arranged to slide between the wheel and casing, and by their means the range of action could be varied from 300 H.P. at 120 revs. per min. to the maximum.
Professor Osborne Reynolds in 1887 patented a water-brake (see _Proc. Inst. C.E._ 99, p. 167), using Froude's turbine to obtain the highly resisting spiral vortices, and arranging passages in the casing for the entry of water at the hub of the wheel and its exit at the circumference. Water enters at E (fig. 5), and finds its way into the interior of the wheel, A, driving the air in front of it through the air-passages K, K. Then following into the pocketed chambers V1, V2, it is caught into the vortex, and finally escapes at the circumference, flowing away at F. The air-ways k, k, in the fixed vanes establish communication between the cores of the vortices and the atmosphere. From {1/5} to 30 H.P. may be measured at 100 revs. per min. by a brake-wheel of this kind 18 in. in diameter. For other speeds the power varies as the cube of the speed. The casing is held from turning by weights hanging on an attached arm. The cocks regulating the water are connected to the casing, so that any tilting automatically regulates the flow, and therefore the thickness of the film in the vortex. In this way the brake may be arranged to maintain a constant torque, not withstanding variation of the speed. In G.I. Alden's brake (see _Trans. Amer. Soc. Eng._ vol. xi.) the resistance is obtained by turning a cast iron disk against the frictional resistance of two thin copper plates, which are held in a casing free to turn upon the shaft, and are so arranged that the pressure between the rubbing surfaces is controlled, and the heat developed by friction carried away, by the regulated flow of water through the casing. The torque required to hold the casing still against the action of the disk measures the torque exerted by the shaft to which the disk is keyed.
_Transmission Dynamometers._--The essential part of many transmission dynamometers is a spring whose deformation indirectly measures the magnitude of the force transmitted through it. For many kinds of spring the change of form is practically proportional to the force, but the relation should always be determined experimentally. General A.J. Morin (see _Notice sur divers appareils dynamometriques_, Paris, 1841), in his classical experiments on traction, arranged his apparatus so that the change in form of the spring was continuously recorded on a sheet of paper drawn under a style. For longer experiments he used a "Compteur" or mechanical integrator, suggested by J.V. Poncelet, from which the work done during a given displacement could be read off directly. This device consists of a roller of radius r, pressed into contact with a disk. The two are carried on a common frame, so arranged that a change in form of the spring causes a relative displacement of the disk and roller, the point of contact moving radially from or towards the centre of the disk. The radial distance x is at any instant proportional to the force acting through the spring. The angular displacement, [theta], of the disk is made proportional to the displacement, s, of the point of application of the force by suitable driving gear. If d[phi] is the angular displacement of the roller corresponding to displacements, d[theta] of the disk, and ds of the point of application of P, a, and C constants, then
xd[theta] a d[phi] = --------- = -- P ds = C.P ds, r r _ /s2 and therefore [phi] = C | P ds; _/s1
that is, the angular displacement of the roller measures the work done during the displacement from s1 to s2. The shaft carrying the roller is connected to a counter so that [phi] may be observed. The angular velocity of the shaft is proportional to the rate of working. Morin's dynamometer is shown in fig. 6. The transmitting spring is made up of two flat bars linked at their ends. Their centres s1, s2, are held respectively by the pieces A, B, which together form a sliding pair. The block A carries the disk D, B carries the roller R and counting gear. The pulley E is driven from an axle of the carriage. In a dynamometer used by F.W. Webb to measure the tractive resistance of trains on the London & North-Western railway, a tractive pull or push compresses two spiral springs by a definite amount, which is recorded to scale by a pencil on a sheet of paper, drawn continuously from a storage drum at the rate of 3 in. per mile, by a roller driven from one of the carriage axles. Thus the diagram shows the tractive force at any instant. A second pencil electrically connected to a clock traces a time line on the diagram with a kick at every thirty seconds. A third pencil traces an observation line in which a kick can be made at will by pressing any one of the electrical pushes placed about the car, and a fourth draws a datum line. The spring of the dynamometer car used by W. Dean on the Great Western railway is made up of thirty flat plates, 7 ft. 6 in. long, 5 in. x 5/8 in. at the centre, spaced by distance pieces nibbed into the plates at the centre and by rollers at the ends. The draw-bar is connected to the buckle, which is carried on rollers, the ends of the spring resting on plates fixed to the under-frame. The gear operating the paper roll is driven from the axle of an independent wheel which is let down into contact with the rail when required. This wheel serves also to measure the distance travelled. A Morin disk and roller integrator is connected with the apparatus, so that the work done during a journey may be read off. Five lines are traced on the diagram.
In spring dynamometers designed to measure a transmitted torque, the mechanical problem of ascertaining the change of form of the spring is complicated by the fact that the spring and the whole apparatus are rotating together. In the Ayrton and Perry transmission dynamometer or spring coupling of this type, the relative angular displacement is proportional to the radius of the circle described by the end of a light lever operated by mechanism between the spring-connected parts. By a device used by W.E. Dalby (_Proc. Inst. C.E._ 1897-1898, p. 132) the change in form of the spring is shown on a fixed indicator, which may be placed in any convenient position. Two equal sprocket wheels Q1, Q2, are fastened, the one to the spring pulley, the other to the shaft. An endless band is placed over them to form two loops, which during rotation remain at the same distance apart, unless relative angular displacement occurs between Q1 and Q2 (fig. 7) due to a change in form of the spring. The change in the distance d is proportional to the change in the torque transmitted from the shaft to the pulley. To measure this, guide pulleys are placed in the loops guided by a geometric slide, the one pulley carrying a scale, and the other an index. A recording drum or integrating apparatus may be arranged on the pulley frames. A quick variation, or a periodic variation of the magnitude of the force or torque transmitted through the springs, tends to set up oscillations, and this tendency increases the nearer the periodic time of the force variation approaches a periodic time of the spring. Such vibrations may be damped out to a considerable extent by the use of a dash-pot, or may be practically prevented by using a relatively stiff spring.
Every part of a machine transmitting force suffers elastic deformation, and the force may be measured indirectly by measuring the deformation. The relation between the two should in all cases be found experimentally. G.A. Hirn (see _Les Pandynamometres_, Paris, 1876) employed this principle to measure the torque transmitted by a shaft. Signor Rosio used a telephonic method to effect the same end, and mechanical, optical and telephonic devices have been utilized by the Rev. F.J. Jervis-Smith. (See _Phil. Mag._ February 1898.)
H. Frahm,[1] during an important investigation on the torsional vibration of propeller shafts, measured the relative angular displacement of two flanges on a propeller shaft, selected as far apart as possible, by means of an electrical device (_Engineering_, 6th of February 1903). These measurements were utilized in combination with appropriate elastic coefficients of the material to find the horse-power transmitted from the engines along the shaft to the propeller. In this way the effective horse-power and also the mechanical efficiency of a number of large marine engines, each of several thousand horse-power, have been determined.
When a belt, in which the maximum and minimum tensions are respectively P and p lb., drives a pulley, the torque exerted is (P - p)r lb. ft., r being the radius of the pulley plus half the thickness of the belt. P and p may be measured directly by leading the belt round two freely hanging guide pulleys, one in the tight, the other in the slack part of the belt, and adjusting loads on them until a stable condition of running is obtained. In W. Froude's belt dynamometer (see _Proc. Inst. M.E._, 1858) (fig. 8) the guide pulleys G1, G2 are carried upon an arm free to turn about the axis O. H is a pulley to guide the approaching and receding parts of the belt to and from the beam in parallel directions. Neglecting friction, the unbalanced torque acting on the beam is 4r{P - p} lb. ft. If a force Q acting at R maintains equilibrium, QR/4 = (P - p)r = T. Q is supplied by a spring, the extensions of which are recorded on a drum driven proportionally to the angular displacement of the driving pulley; thus a work diagram is obtained. In the Farcot form the guide pulleys are attached to separate weighing levers placed horizontally below the apparatus. In a belt dynamometer built for the Franklin Institute from the designs of Tatham, the weighing levers are separate and arranged horizontally at the top of the apparatus. The weighing beam in the Hefner-Alteneck dynamometer is placed transversely to the belt (see _Electrotechnischen Zeitschrift_, 1881, 7). The force Q, usually measured by a spring, required to maintain the beam in its central position is proportional to (P - p). If the angle [theta]1 = [theta]2 = 120 deg., Q = (P - p) neglecting friction.
When a shaft is driven by means of gearing the driving torque is measured by the product of the resultant pressure P acting between the wheel teeth and the radius of the pitch circle of the wheel fixed to the shaft. Fig. 9, which has been reproduced from J. White's _A New Century of Inventions_ (Manchester, 1822), illustrates possibly the earliest application of this principle to dynamometry. The wheel D, keyed to the shaft overcoming the resistance to be measured, is driven from wheel N by two bevel wheels L, L, carried in a loose pulley K. The two shafts, though in a line, are independent. A torque applied to the shaft A can be transmitted to D, neglecting friction, without change only if the central pulley K is held from turning; the torque required to do this is twice the torque transmitted.
The torque acting on the armature of an electric motor is necessarily accompanied by an equal and opposite torque acting on the frame. If, therefore, the motor is mounted on a cradle free to turn about knife-edges, the reacting torque is the only torque tending to turn the cradle when it is in a vertical position, and may therefore be measured by adjusting weights to hold the cradle in a vertical position. The rate at which the motor is transmitting work is then T2[pi]n/550 H.P., where n is the revolutions per second of the armature.
See James Dredge, _Electric Illumination_, vol. ii. (London, 1885); W.W. Beaumont, "Dynamometers and Friction Brakes," _Proc. Inst. C.E._ vol. xcv. (London, 1889); E. Brauer, "Ueber Bremsdynamometer and verwandte Kraftmesser," _Zeitschrift des Vereins deutscher Ingenieure_ (Berlin, 1888); J.J. Flather, _Dynamometers and the Measurement of Power_ (New York, 1893). (W. E. D.)
FOOTNOTE:
[1] H. Frahm, "Neue Untersuchungen ueber die dynamischen Vorgaenge in den Wellenleitungen von Schiffsmaschinen mit besonderer Beruecksichtigung der Resonanzschwingungen," _Zeitschrift des Vereins deutscher Ingenieure_, 31st May 1902.
DYNASTY (Gr. [Greek: dynasteia], sovereignty, the position of a [Greek: dynastes], lord, ruler, from [Greek: dynasthai], to be able, [Greek: dynamis], power), a family or line of rulers, a succession of sovereigns of a country belonging to a single family or tracing their descent to a common ancestor. The term is particularly used in the history of ancient Egypt as a convenient means of arranging the chronology.
DYSART, a royal and police burgh and seaport of Fifeshire, Scotland, on the shore of the Firth of Forth, 2 m. N.E. of Kirkcaldy by the North British railway. Pop. (1901) 3562. It has a quaint old-fashioned appearance, many ancient houses in High Street bearing inscriptions and dates. The public buildings include a town hall, library, cottage hospital, mechanics' institute and memorial hall. Scarcely anything is left of the old chapel dedicated to St Dennis, which for a time was used as a smithy; and of the chapel of St Serf, the patron saint of the burgh, only the tower remains. The chief industries are the manufacture of bed and table linen, towelling and woollen cloth, shipbuilding and flax-spinning. There is a steady export of coal, and the harbour is provided with a wet dock and patent slip. In smuggling days the "canty carles" of Dysart were professed "free traders." In the 15th and 16th centuries the town was a leading seat of the salt industry ("salt to Dysart" was the equivalent of "coals to Newcastle"), but the salt-pans have been abandoned for a considerable period. Nail-making, once famous, is another extinct industry. During the time of the alliance between Scotland and Holland, which was closer in Fifeshire than in other counties, Dysart became known as Little Holland. To the west of the town is Dysart House, the residence of the earl of Rosslyn. With Burntisland and Kinghorn Dysart forms one of the Kirkcaldy district group of parliamentary burghs. The town is mentioned as early as 874 in connexion with a Danish invasion. Its name is said to be a corruption of the Latin _desertum_, "a desert," which was applied to a cave on the seashore occupied by St Serf. In the cave the saint held his famous colloquy with the devil, in which Satan was worsted and contemptuously dismissed. From James V. the town received the rights of a royal burgh. In 1559 it was the headquarters of the Lords of the Congregation, and in 1607 the scene of the meetings of the synod of Fife known as the Three Synods of Dysart. Ravensheugh Castle, on the shore to the west of the town, is the Ravenscraig of Sir Walter Scott's ballad of "Rosabelle."
William Murray, a native of the place, was made earl of Dysart in 1643, and his eldest child and heir, a daughter, Elizabeth, obtained in 1670 a regrant of the title, which passed to the descendants of her first marriage with Sir Lionel Tollemache, Bart., of Helmingham; she married secondly the 1st duke of Lauderdale, but had no children by him, and died in 1698. This countess of Dysart (afterwards duchess of Lauderdale) was a famous beauty of the period, and notorious both for her amours and for her political influence. She was said to have been the mistress of Oliver Cromwell, and also of Lauderdale before her first husband's death, and was a leader at the court of Charles II. Wycherley is supposed to have aimed at her in his Widow Blackacre in the _Plain Dealer_. Her son, Lionel Tollemache (d. 1727), transmitted the earldom to his grandson Lionel (d. 1770), whose sons Lionel (d. 1799) and Wilbraham (d. 1821) succeeded; they died without issue, and their sister Louisa (d. 1840), who married John Manners, an illegitimate son of the second son of the 2nd duke of Rutland, became countess in her own right, being succeeded by her grandson (d. 1878), and his grandson, the 8th earl.
The earldom of Dysart must not be confounded with that of Desart (Irish), created (barony 1733) in 1793, and held in the Cuffe family, who were originally of Creech St Michael, Somerset, the Irish branch dating from Queen Elizabeth's time.
DYSENTERY (from the Gr. prefix [Greek: dys]-, in the sense of "bad," and [Greek: enteron], the intestine), also called "bloody flux," an infectious disease with a local lesion in the form of inflammation and ulceration of the lower portion of the bowels. Although at one time a common disease in Great Britain, dysentery is now very rarely met with there, and is for the most part confined to warm countries, where it is the cause of a large amount of mortality. (For the pathology see DIGESTIVE ORGANS.)
Recently considerable advance has been made in our knowledge of dysentery, and it appears that there are two distinct types of the disease: (1) amoebic dysentery, which is due to the presence of the amoeba histolytica (of Schaudinn) in the intestine; (2) bacillary dysentery, which has as causative agent two separate bacteria, (a) that discovered by Shiga in Japan, (b) that discovered by Flexner in the Philippine Islands. With regard to the bacillary type, at first both organisms were considered to be identical, and the name _bacillus dysenteriae_ was given to them; but later it was shown that these bacilli are different, both in regard to their cultural characteristics and also in that one (Shiga) gives out a soluble toxin, whilst the other has so far resisted all efforts to discover it. Further, the serum of a patient affected with one of the types has a marked agglutinative power on the variety with which he is infected and not on the other.
Clinically, dysentery manifests itself with varying degrees of intensity, and it is often impossible without microscopical examination to determine between the amoebic and bacillary forms. In well-marked cases the following are the chief symptoms. The attack is commonly preceded by certain premonitory indications in the form of general illness, loss of appetite, and some amount of diarrhoea, which gradually increases in severity, and is accompanied with griping pains in the abdomen (tormina). The discharges from the bowels succeed each other with great frequency, and the painful feeling of pressure downwards (tenesmus) becomes so intense that the patient is constantly desiring to defecate. The matters passed from the bowels, which at first resemble those of ordinary diarrhoea, soon change their character, becoming scanty, mucous or slimy, and subsequently mixed with, or consisting wholly of, blood, along with shreds of exudation thrown off from the mucous membrane of the intestine. The evacuations possess a peculiarly offensive odour characteristic of the disease. Although the constitutional disturbance is at first comparatively slight, it increases with the advance of the disease, and febrile symptoms come on attended with urgent thirst and scanty and painful flow of urine. Along with this the nervous depression is very marked, and the state of prostration to which the patient is reduced can scarcely be exceeded. Should no improvement occur death may take place in from one to three weeks, either from repeated losses of blood, or from gradual exhaustion consequent on the continuance of the symptoms, in which case the discharges from the bowels become more offensive and are passed involuntarily.
When, on the other hand, the disease is checked, the signs of improvement are shown in the cessation of the pain, in the evacuations being less frequent and more natural, and in relief from the state of extreme depression. Convalescence is, however, generally slow, and recovery may be imperfect--the disease continuing in a chronic form, which may exist for a variable length of time, giving rise to much suffering, and not unfrequently leading to an ultimately fatal result.
The dysentery poison appears to exert its effects upon the glandular structures of the large intestine, particularly in its lower part. In the milder forms of the disease there is simply a congested or inflamed condition of the mucous membrane, with perhaps some inflammatory exudation on its surface, which is passed off by the discharges from the bowels. But in the more severe forms ulceration of the mucous membrane takes place. Commencing in and around the solitary glands of the large intestine in the form of exudations, these ulcers, small at first, enlarge and run into each other, till a large portion of the bowel may be implicated in the ulcerative process. Should the disease be arrested these ulcers may heal entirely, but occasionally they remain, causing more or less disorganization of the coats of the intestines, as is often found in chronic dysentery. Sometimes, though rarely, the ulcers perforate the intestines, causing rapidly fatal inflammation of the peritoneum, or they may erode a blood vessel and produce violent haemorrhage. Even where they undergo healing they may cause such a stricture of the calibre of the intestinal canal as to give rise to the symptoms of obstruction which ultimately prove fatal. One of the severest complications of the disease is abscess of the liver, usually said to be solitary, and known as tropical abscess of the liver, but probably is more frequently multiple than is usually thought.
_Treatment._--Where the disease is endemic or is prevailing epidemically, it is of great importance to use all preventive measures, and for this purpose the avoidance of all causes likely to precipitate an attack is to be enjoined. Exposure to cold after heat, the use of unripe fruit, and intemperance in eating and drinking should be forbidden; and the utmost care taken as to the quality of the food and drinking water. In houses or hospitals where cases of the disease are under treatment, disinfectants should be freely employed, and the evacuations of the patients removed as speedily as possible, having previously been sterilized in much the same manner as is employed in typhoid fever. In the milder varieties of this complaint, such as those occurring sporadically, and where the symptoms are probably due to matters in the bowels setting up the dysenteric irritation, the employment of diaphoretic medicines is to be recommended, and the administration of such a laxative as castor oil, to which a small quantity of laudanum has been added, will often, by removing the source of the mischief, arrest the attack; but a method of treatment more to be recommended is the use of salines in large doses, such as one drachm of sodium sulphate from four to eight times a day. This treatment may with advantage be combined with the internal administration of ipecacuanha, which still retains its reputation in this disease. Latterly, free irrigation of the bowel with astringents, such as silver nitrate, tannalbin, &c., has been attended with success in those cases which have been able to tolerate the injections. In many instances they cannot be used owing to the extreme degree of irritability of the bowel. The operation of appendicostomy, or bringing the appendix to the surface and using it as the site for the introduction of the irrigating fluid, has been attended with considerable success.
In those cases due to Shiga's bacillus the ideal treatment has been put at our disposal by the preparation of a specific antitoxin; this has been given a trial in several grave epidemics of late, and may be said to be the most satisfactory treatment and offer the greatest hope of recovery. It is also of great use as a prophylactic.
The preparations of morphia are of great value in the symptomatic treatment of the disease. They may be applied externally as fomentations, for the relief of tormina; by rectal injection for the relief of the tenesmus and irritability of the bowel; hypodermically in advanced cases, for the relief of the general distress. In amoebic dysentery, warm injections of quinine _per rectum_ have proved very efficacious, are usually well tolerated, and are not attended with any ill effects. The diet should be restricted, consisting chiefly of soups and farinaceous foods; more especially is this of importance in the chronic form. For the thirst ice may be given by the mouth. Even in the chronic forms, confinement to bed and restriction of diet are the most important elements of the treatment. Removal from the hot climate and unhygienic surroundings must naturally be attended to.
BIBLIOGRAPHY.--Allbutt and Rolleston, _System of Medicine_, vol. ii.