Part 30

I found that the situation of my instrument was not sensibly altered between the 18th and 19th of October; for the transits and the difference of declination of the same stars being observed with it again on the 19th of October, they agreed very well with those that were taken the preceding night. It may therefore be supposed, that the position of the instrument continued the same likewise during the time of the foregoing observations.

The right ascension of the 17th star of Eridanus being 49° 39’ 10" and its declination 5° 55’ 25" south; and the right ascension of _b_ of Eridanus being 73° 59’ 15" and its declination 5° 25’ 10" south; I collected, that when the Comet passed the wire (or horary circle) which was October 17ᵈ 17ʰ 12’ mean time, its right ascension was 182° 34’ 0" and its declination 5° 45’ 35" south.

The last time that I saw the Comet was on the 19th of October in the morning; but it then appeared so faint, that I could not observe its place. Its elongation from the sun was then but about 20 degrees; and from that day to the present it hath always been less; which is the principal reason why it was invisible to us at the time when it was in its perihelion, and hath remained so ever since. The elongation will indeed soon become greater, and yet it is probable that we shall not be able to see the Comet again; because its real distance from the sun will be greater than it was when I first saw it, and it will be also four times further from us than it was at that time.

The Comet kept nearly at the same distance from the earth for ten or twelve days together after I first saw it; but its brightness gradually increased then, because it was going nearer to the sun. Afterwards, when its distance from the earth increased, altho’ it continued to approach the sun, yet its lustre never much exceeded that of stars of the second magnitude, and the tail was scarce to be discerned by the naked eye.

All the forementioned observations were made with a Micrometer in a seven-foot Tube, excepting those of the 3d, 11th, and 17th days of October, which were taken with a curious Sector constructed for such purposes by the late ingenious Mr. George Graham; of which Dr. Smith has given a very exact description in his third book of Optics.

Supposing the Trajectory of this Comet to be parabolic, I collected from the foregoing observations, that its motion round the sun is _direct_, and that it was in its _perihelion_ October the 21st, at 7ʰ 55’ mean (or equated) time at Greenwich. That the inclination of the plane of its Trajectory to the ecliptic is 12° 50’ 20"; the place of the descending Node ♉ 4° 12’ 50"; the place of the Perihelion ♄ 2° 58’ 0"; the distance of the Perihelion from the descending Node 88° 45’ 10"; the Logarithm of the Perihelion distance 9.528328; the Logarithm of the diurnal motion 0.667636.

From these Elements (which are adapted to Dr. Halley’s general Table for the Motion of Comets in parabolic Orbits), I computed the places of this Comet for the respective times of the foregoing observations, as in the following table; which contains likewise the longitudes and latitudes deduced from the observed right ascensions and declinations, and also the differences between the computed and observed places. These differences (no-where exceeding 40") shew, that the elements here set down will be sufficient to enable future astronomers to distinguish this Comet upon another return; but as they do not correspond with the elements of the orbit of any other Comet hitherto taken notice of, we cannot determine at present the period thereof.

Greenwich, 1757.| Comet. Long.| Mean Time. | Observ. | Latit. Observ. ----------------+--------------+-----------------+ _d._ _h._ '| S. ° ’ " | ° ’ " ----------------+--------------+-----------------+ Sept. 12 16 2 | ♊ 29 34 13 | 11 32 16 No. 13 12 37 | ♋ 2 35 34 | 11 12 13 14 14 0 | 6 27 45 | 10 44 3 ----------------+-------------+-----------------+ 17 13 0 | 17 49 40 | 9 3 31 19 15 17 | 26 6 8 | 7 36 49 23 15 57 | ♌ 11 19 18 | 4 33 38 ----------------+-------------+-----------------+ 24 15 21 | 14 44 19 | 3 49 37 28 16 22 | 27 23 43 | 1 3 44 No. 30 16 24 | ♍ 2 45 43 | 0 5 30 So. ----------------+-------------+-----------------+ Octob. 2 16 48 | 7 37 43 | 1 5 50 3 16 45 | 9 51 36 | 1 32 22 4 17 0 | 12 1 4 | 1 56 42 ----------------+-------------+-----------------+ 7 16 54 | 17 51 3 | 2 56 48 8 16 53 | 19 39 45 | 3 13 7 11 16 52 | 24 47 22 | 3 48 49 17 17 12 | ♎ 4 38 58 | 4 15 42 So.

Greenwich, 1757.| | | Diff. | Diff. Mean Time. | Long. Comp. | Latit. Comput. | Long. | Latit. ----------------+----------------+---------------+-------+-------- _d._ _h._ '| S. ° ’ " | ° ’ " | " | " ----------------+----------------+----------------+-------+------- Sept. 12 16 2 | ♊ 29 34 11 | 11 32 20 No. | -2 | +4 13 12 37 | ♋ 2 35 47 | 11 12 11 | +13 | -2 14 14 0 | 6 27 42 | 10 43 43 | -3 | -20 ----------------+----------------+----------------+-------+------- 17 13 0 | 17 50 16 | 9 3 11 |+36 |-20 19 15 17 | 26 5 50 | 7 36 30 |-18 |-19 23 15 57 | ♌ 11 19 4 | 4 33 32 |-14 | -6 ----------------+----------------+----------------+-------+------- 24 15 21 | 14 44 3 | 3 49 39 |-16 | +2 28 16 22 | 27 23 32 | 1 3 52 No. |-11 | +8 30 16 24 | ♍ 2 45 39 | 0 5 17 So. | -4 |-13 ----------------+----------------+----------------+-------+------- Octob. 2 16 48 | 7 37 42 | 1 5 32 | -1 |-18 3 16 45 | 9 51 29 | 1 31 55 | -7 |-27 4 17 0 | 12 0 25 | 1 56 23 | -39 | -19 ----------------+----------------+----------------+-------+------ 7 16 54 | 17 51 6 | 2 56 24 | +3 | -24 8 16 53 | 19 39 33 | 3 12 28 | -12 | -39 11 16 52 | 24 47 47 | 3 49 29 | +25 | +40 17 17 12 | ♎ 4 38 36 | 4 15 2 So. | -22 | -40

LIII. _The Resolution of a General Proposition for Determining the_ horary _Alteration of the Position of the Terrestrial Equator, from the Attraction of the Sun and Moon: With some Remarks on the Solutions given by other Authors to that difficult and important Problem. By Mr._ Tho. Simpson, _F.R.S._

[Read Dec. 22, 1757.]

SINCE the time, that that excellent Astronomer, my much honoured friend Dr. Bradley, published his observations and discoveries concerning the inequalities of the precession of the equinox, and of the obliquity of the ecliptic, depending on the position of the lunar nodes, mathematicians, in different parts of Europe, have set themselves diligently to compute, from physical principles, the effects produced by the sun and moon, in the position of the terrestrial equator; and to examine whether these effects do really correspond with the observations.

Two papers on this subject have already appeared in the Philosophical Transactions; in which the authors have shewn evident marks of skill and penetration. There is, nevertheless, one part of the subject, that seems to have been passed over without a due degree of attention, as well by both those gentlemen, as by Sir Isaac Newton himself.

This part, which, upon account of physical difficulties, is indeed somewhat slippery and perplexing, I shall make the principal subject of this essay.

GENERAL PROPOSITION.

_Supposing an homogeneous sphere_ OABCD (Fig. 1.) _revolving uniformly about its centre, to be acted on at the extremity_ A _of the radius_ OA, _in a direction_ AL _perpendicular to the plane of the equator_ ABCD, _and parallel to the axis of rotation_ Pp, _by a given force, tending to generate a new motion of rotation at right angles to the former; It is proposed to determine the change, that will arise in the direction of the rotation in consequence of the said force._

Let _F_ denote the given force, whereby the motion about the axis P_p_ is disturbed, supposing _f_ to represent the centrifugal force of a small particle of matter in the circumference of the equator, arising from the sphere’s rotation; and let the whole number of such particles, or the content of the sphere, be denoted by _c_: let also the momentum of rotation of the whole sphere, or of all the particles, be supposed, in proportion to the momentum of an equal number of particles, revolving at the distance OA of the remotest point A, as _n_ is to _unity_.

It is well known, that the centripetal force, whereby any body is made to revolve in the circumference of a circle, is such, as is sufficient to generate all the motion in the body, in a time equal to _that_, wherein the body describes an arch of the circumference, equal in length to the radius. Therefore, if we here take the arch AR = OA, and assume _m_ to express the time, in which that arch would be uniformly described by the point A, the _motion_ of a particle of matter at A (whose central force is represented by _f_) will be equal to _that_, which might be uniformly generated by the force _f_, in the time _m_; and the motion of as many particles (revolving, all, at the same distance) as are expressed by _cn_ (which, by hypothesis, is equal to the momentum of the whole body), will, consequently, be equal to the momentum, that might be generated by the force _f_ × _cn_, in the same time _m_. Whence it appears, that the momentum of the whole body about its axe P_p_ is in proportion to the momentum generated in a given particle of time _m’_, by the given force _F_ in the direction AL, as _ncf_ × _m_ is to _F_ × _m’_, or, as _unity_ to (_F_/_ncf_) × (_m’_/_m_) (because the quantities of motion produced by unequal forces, in unequal times, are in the ratio of the forces and of the times, conjunctly). Let, therefore, AL be taken in proportion to AM, as (_F_/_ncf_) × (_m’_/_m_) is to _unity_ (supposing AM to be a tangent to the circle ABCD in A), and let the parallelogram AMNL be compleated; drawing also the diagonal AN; then, by the composition of forces, the angle NAM (whose tangent, to the radius OA, is expressed by OA × (_F_/_ncf_) × (_m’_/_m_)) will be the change of the direction of the rotation, at the end of the aforesaid time (_m’_). But, this angle being exceeding small, the tangent may be taken to represent the measure of the angle itself; and, if Z be assumed to represent the arch described by A, in the same time (_m’_) about the center O, we shall also have (_m’_/_m_) = (Z/AR) = (Z/AO), and consequently OA × (_F_/_ncf_) x (_m_/_m’_) = Z × (_F_/_ncf_). From whence it appears, that the angle expressing the change of the direction of the rotation, during any small particle of time, will be in proportion to the angle described about the axe of rotation in the same time, as _F_/_ncf_ is to _unity_. _Q.E.I._

Altho’, in the preceding proposition, the body is supposed to be a perfect sphere, the solution, nevertheless, holds equally true in every other species of figures, as is manifest from the investigation. It is true, indeed, that the value of _n_ will not be the same in these cases, even supposing those of _c_, _f_ and _F_ to remain unchanged; except in the spheroid only, where, as well as in the sphere, _n_ will be = ⅖; the momentum of any spheroid about its axis being 2-5ths of the momentum of an equal quantity of matter placed in the circumference of the equator, as is very easy to demonstrate.

But to shew now the use and application of the general proportion here derived, in determining the regress of the equinoctial points of the terrestrial spheroid, let AE_a_F (_Fig. 2._) be the equator, and P_p_ the axis of the spheroid: also let HECF represent the plane of the ecliptic, S the place of the sun, and HAPNH the plane of the sun’s declination, making right-angles with the plane of the equator AE_a_F: then, if AK be supposed parallel, and OKM perpendicular, to OS, and there be assumed _T_ and _t_ to express the respective times of the annual and diurnal revolutions of the earth, it will appear (from the _Principia_, B. III. prop. xxv.) that the force, with which a particle of matter at A tends to recede from the line OM in consequence of the sun’s attraction, will be expressed by (_3tt_/_TT_) × (AK/OA) × _f_; _f_ denoting the centrifugal force of the same particle, arising from the diurnal rotation. Hence, by the resolution of forces, (_3tt_/_TT_) × (AK/OA) × (OK/OA) × _f_ will be the effect of that particle, in a direction perpendicular to OA, to turn the earth about its center O.

But it is demonstrated by Sir Isaac Newton, and by other authors, that the force of all the particles, or of all the matter in the whole spheroid AP _ap_, to turn _it_ about its center, is equal to ⅕th of the force of a quantity of matter, placed at A, equal to the excess of the matter in the whole spheroid above _that_ in the inscribed sphere, whose axis is P_p_. Now this excess (assuming the ratio of π to 1, to express _that_ of the area of a circle to the square of the radius) will be truly represented by (4π/3) × OP × (OA² - OP²); and, consequently, the force of all the matter in the whole earth, by (_3tt_/_TT_) × (AK/OA) × (OK/OA) × (4π/15) × OP × (OA²- OP²). Let, therefore, this quantity be now substituted for _F_, in the general formula _F_/_ncf_, writing, at the same time, (4π/3) × OA² × OP, and ⅖, in the place of their equals _c_ and _n_; by which means we have (here) (_F_/_ncf_) = (_3tt_/_2TT_) × ((OA² - OP²)/OA²) × ((AK × OK)/OA²). Put the given quantity (_3tt_/_2TT_) × ((OA² - OP²)/OA²) = _k_; and let the angle EA_e_ represent the horary alteration of the position of the terrestrial equator, arising from the force _F_ (here determined), and let the arch E_e_ be the regress of the equinoctial point E, corresponding thereto: then, in the triangle EA_e_ (considered as spherical) it will be sin. _e_ ∶ sin. AE (∷ sin. EA_e_: sin. E_e_) ∷ EA_e_ ∶ E_e_ (= (sin. AE x EA_e_)/sin. E) = _k_ × (sin. AE/sin. E) × ((AK × OK)/OA²) = _k_ × ((sin. AE × cos. AH × sin. AH)/sin. E). But in the triangle EHA, right-angled at A (where HA is supposed to represent the sun’s declination, AE his right ascension, and HE his distance from the equinoctial point E[207]) we have (_per spherics_)

sin. AE ∶ 1 (rad.) ∷ co-t. E ∶ co-t. AH, (sin. AH)² ∶ (sin. EH)² ∷ (sin. E)² ∶ 1² (rad.²)

From whence we get, sin. AE × co-t. AH × (sin. AH)² = (sin. EH)² × co-t. E × (sin. E)². But co-t. AH × sin. AH = co-s. AH × 1 (rad.), and co-t. E × sin. E = co-s. E × 1 (rad.): therefore sin. AE × co-s. AH × sin. AH = (sin. EH)² × co-s. E × sin. E; and, consequently, _k_ × (sin. AE × co-s. AH × sin. AH)/sin. E = _k_ × co-s. E × (sin. EH)² (= E_e_).

Let, now, the sun’s longitude EH be denoted by Z (considered as a flowing quantity); then, (sin. Z)² being = ½-½ co-s. 2 Z, we shall have _k_ × co-s. E × (sin. EH)² = ½_k_ × co-s. E × 1-co-s. 2 Z. But the angle described about the axe of rotation P_p_, in the time that the sun’s longitude is augmented by the particle Ż, will be = (_T/t_) × Ż. Therefore (by the general proposition) we have, as 1: ½_k_ × co-s. E × 1-co-s. 2 Z ∷ (_T/t_) × Ż : ½_k_ × (_T/t_) × co-s. E × Ż - Ż co-s. 2 Z, the true regress of the equinoctial point E, during that time: whose fluent, ½_k_ × (_T_/_t_) × co-s. E × (Z- ½ sin. 2 Z), will consequently be the total regress of the point E, in the time that the sun, by his apparent motion, describes the arch HE or Z; which, on the sun’s arrival at the solstice, becomes barely = ½_k_ × (_T_/_t_) × co-s. E × an arch of 90°: the quadruple whereof, or ½_k_ × (_T_/_t_) × co-s. E × 360° (= (3_t_/4_T_) × ((OA²-OP²)/OA²) × co-s. E × 360°) is therefore the whole annual precession of the equinox caused by the sun. This, in numbers (taking OP/OA = 229/230) comes out (3/(4 × 366¼)) × (2/230½) × 0.917176 × 360° = 21´´ 6´´´.

The very ingenious M. Silvabelle, in his essay on this subject, inserted in the 48th volume of the Philosophical Transactions, makes the quantity of the annual precession of the equinox, caused by the sun, to be the half, only, of what is here determined. But this gentleman appears to have fallen into a twofold mistake. First, in finding the _momenta of rotation_ of the terrestrial spheroid, and of a very slender ring, at the equator thereof; which _momenta_ he refers to an axis perpendicular to the plane of the sun’s declination, instead of the proper axe of rotation, standing at right angles to the plane of the equator. The difference, indeed, arising from thence, with respect to the spheroid (by reason of its near approach to a sphere) will be inconsiderable; but, in the ring, the case will be quite otherwise; the equinoctial points thereof being made to recede just twice as fast as they ought to do. This may seem the more strange, if regard be had to the conclusions, relating to the nodes of a satellite, derived from this very assumption. But, that these conclusions are true, is owing to a second, or subsequent mistake, at Art. 27; where the measure of the sun’s force is taken the half, only, of the true value; by means whereof the motion of the equinoctial points of the ring is reduced to its proper quantity, and the motion of the equinoctial points of the terrestrial spheroid, to the half of what it ought to be.

That expert geometrician M. Cha. Walmsley, in his Essay on the Precession of the Equinox, printed in the last volume of the Philosophical Transactions, has judiciously avoided all mistakes of this last kind, respecting the sun’s force, by pursuing the method, pointed out by Sir Isaac Newton; but, in determining the effect of that force, has fallen into others, not less considerable than those above adverted to.

In his third Lemma, the momentum of the whole Earth, about its diameter, is computed on a supposition, that the momentum or force of each particle is proportional to its distance from the axis of motion, or barely as the quantity of motion in such particle, considered abstractedly. No regard is, therefore, had to the lengths of the unequal levers, whereby the particles are supposed to receive and communicate their motion: which, without doubt, ought to have been included in the consideration.

In his first proposition, he determines, in a very ingenious and concise manner, the true annual motion of the nodes of a ring (or of a single satellite) at the earth’s equator, revolving with the earth itself, about its center, in the time of one siderial day. This motion he finds to be = (3co-s. 23° 29´/4 rad.) × (⅟366¼) × 360°. Then, in order to infer from thence, the motion of the equinoctial points of the earth itself, he, first, diminishes that quantity, in the ratio of 2 to 5: Because (as is demonstrated by Sir Isaac Newton in his 2d Lemma) the whole force of all the particles situated without the surface of a sphere, inscribed in the spheroid, to turn the body about its center, will be only 2-5ths of the force of an equal number of particles uniformly disposed round the whole circumference of the equator, in the fashion of a ring. The quantity ((3co-s. 23° 29´/4 rad.) × ⅖ × (⅟366¼) × 360°) thus arising, will, therefore, express the true motion of the equinoctial points of a ring, equal in quantity of matter to the excess of the whole earth above the inscribed sphere, when the force whereby the ring tends to turn about its diameter is supposed equal to the force whereby the earth itself tends to turn about the same diameter, in consequence of the sun’s attraction. Thus far our author agrees with Sir Isaac Newton; but, in deriving from hence the motion of the equinoctial points of the earth itself, he differs from him; and, in the corollary to his third Lemma, assigns the reasons, why he thinks Sir Isaac Newton, in this particular, has _wandered a little from the truth_. Instead of diminishing the quantity above exhibited (as Sir Isaac has done) in the ratio of all the motion in the ring to the motion in the whole earth, he diminishes it in the ratio of the motion of all the matter above the surface of the inscribed sphere to the motion of the whole earth: which matter, tho’ equal to that of the ring, has nevertheless a different momentum, arising from the different situation of the particles in respect to the axis of motion.

But since the aforesaid quantity, from whence the motion of the earth’s equinox is derived, as well by this gentleman, as by Sir Isaac Newton, expresses truly the annual regress of the equinoctial points of the ring (and not of the hollow figure formed by the said matter, which is greater, in the ratio of 5 to 4) it seems, at least, as reasonable to suppose, that the said quantity, to obtain from thence the true regress of the equinoctial points of the earth, ought to be diminished in the former of the two ratios above specified, as that it should be diminished in the latter. But, indeed, both these ways are defective, even supposing the momenta to have been truly computed; the ratio, that ought to be used here, being that of the momenta of the ring and earth about the proper axe of rotation of the two figures, standing at right-angles to the plane of the ring and of the equator. Now this ratio, by a very easy computation, is found to be as 230²-229² to ⅖ of 230²; whence the quantity sought comes out = (3co-s. 23° 29´/4 rad.) × (⅟366¼) × (230²-229²)/230² × 360° = 21´´ 6´´´: which is the same that we before found it to be, and the double of what this author makes it.

What has been said hitherto, relates to that part of the motion only, arising from the force of the sun. It will be but justice to observe here, that the effect of the moon, and the inequalities depending on the position of her nodes, are truly assigned by both the gentlemen above-named; the ratio of the diameters of the earth, and the density of the moon being so assumed, as to give the maxima of those inequalities, such as the observations require: in consequence whereof, and from the law of the increase and decrease (which is rightly determined by theory, tho’ the absolute quantity is not) a true solution, in every other circumstance, is obtained.

The freedom, with which I have expressed myself, and the liberty I have here taken, to animadvert on the works of men, who, in many places, have given incontestible proofs of skill and genius, may, I fear, stand in need of some apology. ’Tis possible I may be thought too peremptory. Indeed, I might have delivered my sentiments with more caution and address: but, had not I imagined myself quite clear in what has been advanced, from a multitude of concurrent reasons, I should have thought it too great a presumption to have said any thing at all here, on this subject. The great regard I have for this Society, of which I have the honour to be a member, will, I hope, be considered as the motive for my having attempted to rectify some oversights, that have occurred in the works of this learned body.

LIV. _Remarks upon the Heat of the Air in_ July 1757. _in an Extract of a Letter from_ John Huxham, _M.D. F.R.S. to_ William Watson, _M.D. F.R.S. dated at_ Plymouth _19th of that Month. With additional Remarks by Dr._ Watson.

[Read Dec. 22, 1757.]