Chapter 3

A modification of this train better suited for practical use is shown in Fig. 37, in which the sun-wheel, instead of the planet, is annular, and the latter is carried by the two eccentrics, E, E, whose throw is equal to the difference between the diameters of the two pitch circles; these eccentrics must, of course, be driven in the same direction and at equal speeds, like the cranks in Fig. 36.

A curious arrangement of pin-gearing is shown in Fig. 38: in this case the diameter of the pinion is half that of the annular wheel, and the latter being the driver, the elementary hypocycloidal faces of its teeth are diameters of its pitch circle; the derived working tooth-outlines for pins of sensible diameter are parallels to these diameters, of which fact advantage is taken to make the pins turn in blocks which slide in straight slots as shown. The formula is the same as that for Fig. 36, viz.:

V' = v'(1 - f/F),

which, since f = 2F, reduces to V' = -v'.

Of the same general nature is the combination known as the "Epicycloidal Multiplying Gear" of Elihu Galloway, represented in Fig. 39. Upon examination it will be seen, although we are not aware that attention has previously been called to the fact, that this differs from the ordinary forms of "pin gearing" only in this particular, viz., that the elementary tooth of the driver consists of a complete branch, instead of a comparatively small part of the hypocycloid traced by rolling the smaller pitch-circle within the larger. It is self-evident that the hypocycloid must return into itself at the point of beginning, without crossing: each branch, then, must subtend an aliquot part of the circumference, and can be traced also by another and a smaller describing circle, whose diameter therefore must be an aliquot part of the diameter of the outer pitch-circle; and since this last must be equal to the sum of the diameters of the two describing circles, it follows that the radii of the pitch circles must be to each other in the ratio of two successive integers; and this is also the ratio of the number of pins to that of the epicycloidal branches.

Thus in Fig. 39, the diameters of the two pitch circles are to each other as 4 to 5; the hypocycloid has 5 branches, and 4 pins are used. These pins must in practice have a sensible diameter, and in order to reduce the friction this diameter is made large, and the pins themselves are in the form of rollers. The original hypocycloid is shown in dotted line, the working curve being at a constant normal distance from it equal to the radius of the roller; this forms a sort of frame or yoke, which is hung upon cranks as in Figs. 36 and 38. The expression for the velocity ratio is the same as in the preceding case:

V¹ = v'(1 - f/F); which in Fig. 39 gives

V¹ = v'(1 - 5/4)= -¼v':

the planet wheel, or epicycloidal yoke, then, has the higher speed, so that if it be desired to "gear up," and drive the propeller faster than the engine goes (and this, we believe, was the purpose of the inventor), the pin-wheel must be made the driver; which is the reverse of advantageous in respect to the relative amounts of approaching and receding action.

In Figs. 40 and 41 are given the skeletons of Galloway's device for ratios of 3:4 and 2:3 respectively, the former having four branches and three pins, the latter three branches and two pins. Following the analogy, it would seem that the next step should be to employ two branches with only one pin; but the rectilinear hypocycloid of Fig. 38 is a complete diameter, and the second branch is identical with the first; the straight tooth, then, could theoretically drive the pin half way round, but upon its reaching the center of the outer wheel, the driving action would cease: this renders it necessary to employ two pins and two slots, but it is not essential that the latter should be perpendicular to each other.

In these last arrangements, the forms of the parts are so different from those of ordinary wheels, that the true nature of the combinations is at least partially disguised. But it may be still more completely hidden, as for instance in the common elliptic trammel, Fig. 42. The slotted cross is here fixed, and the pins, R and P, sliding respectively in the vertical and horizontal lines, control the motion of the bar which carries the pencil, S. At first glance there would seem to be nothing here resembling wheel works. But if we describe a circle upon R P as a diameter, its circumference will always pass through C, because R C P is a right angle, and the instantaneous axis of the bar being at the intersection O of a vertical line through P, with a horizontal line through R, will also lie upon this circumference. Again, since O is diametrically opposite to C, we have C O = R P, whence a circle about center C with radius R P will also pass through O, which therefore is the point of contact of these two circles. It will now be seen that the motion of the bar is the same as though carried by the inner circle while rolling within the outer one, the latter being fixed; the points P and R describing the diameters L M and K N, the point D a circle, and S an ellipse; C D being the train-arm. The distance R P being always the diameter of one circle and the radius of the other, the sizes of the wheels can be in effect varied by altering that distance.

Thus we see that this combination is virtually the same in its action as the one shown in Fig. 43, known as Suardi's Geometrical Pen. In this particular case the diameter of _a_ is half of that of A; these wheels are connected by the idler, E, which merely reverses the direction without affecting the velocity of _a's_ rotation. The working train arm is jointed so as to pivot about the axis of E, and may be clamped at any angle within its range, thus changing the length of the virtual train arm, C D. The bar being fixed to _a_, then, moves as though carried by the wheel, _a¹_, rolling within A¹; the radius of _a¹_ being C D, and that of A¹ twice as great.

In either instrument, the semi-major axis C X is equal to S R, and the semi-minor axis to S P.

The _ellipse_, then, is described by these arrangements because it is a special form of the epitrochoid; and various other epitrochoids may be traced with Suardi's pen by substituting other wheels, with different numbers of teeth, for a in Fig. 43.

Another disguised planetary arrangement is found in Oldham's coupling, Fig. 44. The two sections of shafting, A and B, have each a flange or collar forged or keyed upon them; and in each flange is planed a transverse groove. A third piece, C, equal in diameter to the flanges, is provided on each side with a tongue, fitted to slide in one of the grooves, and these tongues are at right angles to each other. The axes of A and B must be parallel, but need not coincide; and the result of this connection is that the two shafts will turn in the same direction at the same rate.

The fact that C in this arrangement is in reality a planetary wheel, will be perceived by the aid of the diagram, Fig. 45. Let C D be two pieces rotating about fixed parallel axes, each having a groove in which slides freely one of the arms, A C, A D, which are rigidly secured to each other at right angles.

The point C of the upper arm can at the instant move only in the direction C A; and the point D of the lower arm only in the direction A D, at the same instant; the instantaneous axis is therefore at the intersection, K, of perpendiculars to A C and A D, at the points C and D. C A D K being then a rectangle, A K and C D will be two diameters of a circle whose center, O, bisects C D; and K will also be the point of contact between this circle and another whose center is A, and radius A K = C D. If then we extend the arms so as to form the cross, P K, M N, and suppose this to be carried by the outer circle, _f_, rolling upon the inner one, F, its motion will be the same as that determined by the pieces, C D; and such a cross is identical with that formed by the tongues on the coupling-piece, C, of Fig. 44.

A O is the virtual train-arm; let the center, A, of the cross move to the position B, then since the angles A O B at the center, and A C B in the circumference, stand on the same arc, A B, the former is double the latter, showing that the cross revolves twice round the center O during each rotation of C; and since A C B = A D B, C and D rotate with equal velocities, and these rotations and the revolution about O have the same direction. While revolving, the cross rotates about its traveling center, A, in the opposite direction, the contact between the two circles being internal, and at a rate equal to that of the rotations of C and D, because the velocities of the axial and the orbital motion are to each other as _f_ is to F, that is to say, as 1 is to 2. Since in the course of the revolution the points P and K must each coincide with C, and the points M and N with D, it follows that each tongue in Fig. 44 must slide in its groove a distance equal to twice that between the axes of the shafts.

Another example of a disguised planetary train is shown in Fig. 46. Let C be the center about which the train arm, T, revolves, and suppose it required that the distant shaft, B, carried by T, shall turn once backward for each forward revolution of the arm. E is a fixed eccentric of any convenient diameter, in the upper side of which is a pin, D. On the shaft, B, is keyed a crank, B G, equal in length to C D; and at any convenient point, H, on B C, or its prolongation, another crank, H F, equal also to C D, is provided with a bearing in the train-arm. The three crank pins, F, D, G, are connected by a rod, like the parallel rod of a locomotive; F D, D G, being respectively equal to H C, C B. Then, as the train-arm revolves, the three cranks must remain parallel to each other; but C D being fixed, the cranks, H F and B G, will remain always parallel to their original positions, thus receiving the required motion of circular translation.

The result then is the same as though the periphery of E were formed into a fixed spurwheel, A, and another, _a_, of the same size, secured on a shaft, B, the two being connected by the three equal wheels, L, M, N. It need hardly be stated that instead of the eccentric, E, a stationary crank similar and equal to B G may be used, should it be found better suited to the circumstances of the case.

It is possible also to apply the planetary principle to mechanism composed partially of racks; in fact, a rack is merely a wheel of prodigious size--the limiting case, just as a right line is a circle of infinite radius. A very neat application of this principle is found in Villa's Pantograph, of which a full description and illustration was given in SCIENTIFIC AMERICAN SUPPLEMENT, No. 424; the racks, moving side by side, are the sun-wheels, and the planet-wheels are the pinions, carried by the traveling socket, by which the motion of one rack is transmitted to the other.

Thus far attention has been called only to combinations of circular wheels. In these the velocity ratios are constant, if we except the cases in which two independent trains converge, the two sun-wheels, or one of them and the train-arm, being driven separately--and even in those, a variable motion of the ultimate follower is obtained only by varying the speed of one or both drivers. It is not, however, necessary to employ circular wheels exclusively or even at all; wheels of other forms are capable of acting together in the relation of sun and planet, and in this way a varying velocity ratio may be produced even with a fixed sun-wheel and a single driver. We have not found, in the works of any previous writer, any intimation that noncircular wheels have ever been thus combined; and we propose in the following article to illustrate some curious results which may be thus obtained.

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THE FALLACY OF THE PRESENT THEORY OF SOUND.

Dr. H.A. Mott recently delivered a lecture before the New York Academy of Sciences, in Columbia College, on the Fallacy of the Present Theory of Sound.

He commenced his lecture by stating that "the object of science was not to find out what we like or what we dislike; the object of science was truth." He then said that, as Galileo stated a hypothesis should be judged by the weight of facts and the force of mathematical deductions, he claimed the theory of sound should be so examined, and not allowed to exist as a true theory simply because it is sustained by a long line of scientific names; as too many theories had been overthrown to warrant the acceptance of any one authority unless they had been thoroughly tested. Dr. Mott stated that Dr. Wilford Hall was the first to attack the theory of sound and show its fallaciousness, and that many other scientists besides himself had agreed with Dr. Hall in his arguments and had advanced additional arguments and experiments to establish this fact. Dr. Mott first gave a very elaborate and still at the same time condensed statement of the current theory of sound as propounded by such men as Helmholtz, Tyndall, Lord Rayleigh, Mayer, Rood, Sir Wm. Thomson, and others, and closed this section of the paper with the remarks made by Tyndall: "Assuredly no question of science ever stood so much in need of revision as this of the transmission of sound through the atmosphere. Slowly but surely we mastered the question, and the further we advance, the more plainly it appeared that our reputed knowledge regarding it was erroneous from beginning to end."

Dr. Mott then took up the other side of the question, and treated the same under the following heads:

1. Agitation of the air. 2. Mobility of the atmosphere. 3. Resonance. 4. Heat and velocity of the supposed sound waves. 5. Decrease in loudness of sound. 6. The physical strength of the locust. 7. The barometric theory of Sir Wm. Thomson. 8. Elasticity and density of the air. 9. Interference and beats. 10. The membrana tympani and the corti arches.

Under the first head Dr. Mott stated that all experiments and photographs made to establish the existence of sound waves simply referred to the necessary agitation of the air accompanying any disturbance, such as would of necessity be produced by a vibrating body, and had nothing to do directly with sound. He stated that in the Edison telephone, sound was converted directly into electricity without vibrating any diaphragm at all, as attested to by Edison himself. Speaking of the mobility of the air, he said the particles were free to slip around and not practically be pushed at all, and that the greatest distance a steam whistle could affect the air would not exceed 30 feet, and the waves would not travel more than 4 or 5 feet a second, while sound travels 1,120 feet a second. Under heat and velocity of sound waves, Dr. Mott stated that Newton found by calculating the exact relative density and elasticity of air that sound should travel only 916 feet a second, while it was known to travel 1,120 feet a second.

Laplace, by a heat and cold theory, tried to account for the 174 feet, and supposed that in the condensed portion of a sound wave heat was generated, and in the rarefied portion cold was produced; the heat augmenting the elasticity and therefore the sound waves, and the cold produced neutralizing the heat, thus kept the atmosphere at a constant temperature. Dr. Mott stated that when Newton first pointed out this discrepancy of 174 feet, the theory should have been dropped at once, and later on he showed the consequences of Laplace's heat and cold theory.

The great argument of the evening, and the one to which he attached the most importance, was that all scientists have spoken of the swift movement of the tuning fork, while in fact it moved 25,000 times slower than the hour hand of a clock and 300,000,000 times slower than any clock pendulum ever constructed.

Since a pendulum cannot, according to the high authorities, produce sonorous air waves on account of its slow movement, Dr. Mott asks some one to enlighten him how a prong of a tuning fork going 300,000,000 times slower could be able to produce them. He then showed that there was not the slightest similarity between the theoretical sound waves and water waves, and still they are spoken of as "precisely similar" and "essentially identical," and "move in exactly the same way." Considerable merriment was occasioned when Dr. Mott showed what a locust stridulating in the air would be called upon to do if the present theory of sound were correct. He stated that a locust not weighing more than half a pennyweight, and that could not move an ounce weight, was supposed capable of setting 4 cubic miles of atmosphere into vibration, weighing 120,000,000 tons, so that it would be displaced 440 times in one second, and any portion of the air could bend the human tympanic membrane once in and once out 440 times in one second; and that 40,000,000 people, nearly the whole population of the United States, could have their 5,000 pounds of tympanic membrane thus shaken by an insect that could not move an ounce weight to save its life; and that the 231,222 pounds of tympanic membrane of the entire population of the earth, amounting to 1,350,000,000, who could conveniently stand in 11¼ square miles, would be affected the same way by 34 locusts stridulating in the air. According to the barometric theory of Sir William Thomson, he showed that a locust would have to add 60,000,000 pounds to the weight of the atmosphere.

Under elasticity and density he stated that elasticity was a mere property of a body, and could not add one grain of force to that exercised by the locust, so as to assist it in performing such wonderful feats. Under interference he showed that the law of interference is fallacious; that no such thing occurs; and that in the experiment with the siren to show such fact, the octave is produced which of necessity ought to be when the number of orifices are alternately doubled, and the same effect would be produced with one disk with double the number of holes. Under the last head of his paper Dr. Mott proved that the membrana tympani was not necessary for good hearing, that in fact when it was punctured, a deaf man could in many cases be made to hear, and in fact it improved the hearing in general; the only reason why the tympanic membrane was not punctured oftener was that dust, heat, and cold were apt to injure the middle ear.

In closing his paper Dr. Mott said that he would risk the fallacy of the current theory of sound on the argument advanced relating to the impossibility of the slow motion of a tuning fork to produce sonorous waves, and stated that he would retire if any one could show the fallacy of the argument; but if not, the wave theory must be abandoned as absurd and fallacious, as was the Ptolemaic system of astronomy, which was handed down from age to age until Copernicus and his aide de camp Galileo gave to the world a better system.

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THE ATTOCK BRIDGE.

We give illustrations from _Engineering_ of a bridge recently constructed across the Indus River at Attock, for the Punjaub Northern State Railway. This bridge, which was opened on May 24, 1883, was erected under the direction of Mr. F.L. O'Callaghan, engineer in chief, Mr. H. Johnson acting as executive engineer, and Messrs. R.W. Egerton and H. Savary as assistants.

The principal spans cover a length of about 1,150 feet. It will be seen from the diagram that there is a difference of nearly 100 feet in the levels of high and low water.

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THE ELASTICITY OF METALS.

M. Tresca has contributed to the _Comptes Rendus_ some observations on the effect of hammering, and the variation of the limit of elasticity of metals and materials used in the arts.

He says that hitherto, in considering the deformation of solids under strain, two distinct periods, relative to their mechanical properties, have alone been recognized. These periods are of course the elastic limit and the breaking point. In the course of M. Tresca's own experiments, however, he has found it necessary to consider, at the end of the period of alteration of elasticity, a third state, geometrically defined and describable as a period of fluidity, corresponding to the possibility of a continuous deformation under the constant action of the same strain. This particular condition is only realized with very malleable or plastic bodies; and it may even be regarded as characteristic of such bodies, since its absence is noticeable in all non-malleable or fragile bodies, which break without being deformed. It is already known that the period of altered elasticity for hard or tempered steel is much less than for iron. In 1871 the author showed that steel or iron rails that had acquired a permanent set were at the same time perfectly elastic up to the limit of the load which they had already borne. With certain bars the same result was renewed five times in succession; and thus their period of perfect elasticity could be successively extended, while the coefficient of elasticity did not appear to sustain any appreciable modification. This process of repeated straining, when there is an absence of a certain hammering effect, renders malleable bodies somewhat similar to those which are not malleable and brittle. There is an indication here of another argument against the testing of steam boilers by exaggerated pressures before use, which process has the effect of rendering the plates more brittle and liable to sudden rupture.

M. Tresca also protests against the elongation of metals under breaking strain tests being stated as a percentage of the length. The elongation is in all cases, chiefly local; and is therefore the same for a test piece 12 inches or 8 inches long, being confined to the immediate vicinity of the point of rupture. The indication of elasticity should rather be sought for in the reduction of the area of the bar at the point of rupture. This portion of the bar is otherwise remarkable for having lost its original condition. It is condensed in a remarkable manner, and has almost completely lost its malleability. The final rupture, therefore, is that of a brittle zone of the metal, of the same character that may be produced by hammering. If a test bar, strained almost to the verge of rupture, be annealed, it will stretch yet further before breaking; and, indeed, by successive annealings and stretchings, may be excessively modified in its proportions.

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THE HARRINGTON ROTARY ENGINE.