« PreviousContinue »
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, by which 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 in which 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 fx cn, in the same time m. Whence it appears, that the momentum of the whole body about its axe pp, 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 rx m', or,
as unity to × (because the quantities of motion produced by unequal forces, in uneqnal times, are in the ratio of the forces and of the times conjunctly). Let therefore AL be taken in proportion to AM, as is to unity (supposing ncf M 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 XXncf will be the change of the direction of the rotation, at the end of the aforesaid time m'. But this angle being exceedingly 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 centre o, we shall also have
=ZX ncf. Whence it ap
pears, 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 is to unity. is to unity. Q. E. I.
Though in the preceding proposition the body is supposed to be a perfect sphere, yet the solution 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 ; the momentum of any spheroid about its axis being two-fifths of the mo
mentum of an equal quantity of matter placed in the circumference of the equator, as is very easy to demonstrate.
But to show now the use and application of the general proportion here derived, in determining the regress of the equinoctial points of the terrestrial spheroid, let AEar (fig. 2) be the equator, and pp 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 AEaF: then, if AK be supposed parallel, and OKм perpendicular, to os, and there be as
sumed T and t to express the respective times of the annual and diurnal revolu→ tions of the earth, it will appear (from the Principia, b. iii. prop. 25) that the force with which a particle of matter at a tends to recede from the line oм, in
consequence of the sun's attraction, will be expressed by × ×ƒ; ƒ denoting the centrifugal force of the same particle arising from the diurnal rotation.
Hence, by the resolution of forces, × × × ƒ will be the effect of
that particle, in a direction perpendicular to oa, to turn the earth about its
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 APap, to turn it about its centre, is equal to 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 pp. 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 OP2); and consequently the force of all the
matter in the whole earth, by × × × 15 × or × (oa2— or2). Let
therefore this quantity be now substituted for F, in the general formula
× n; writing also OA2 X OP, and, instead of their equals c and #; by which
= k; and let the angle EAe represent the horary alteration of the position of the terrestrial equator, arising from the force F here determined, and let the arch Ee be the corresponding regress of the equinoctial point E: then, in the triangle EAE (considered as spherical) it will be sin. e: sin. AE
(:: sin. EAe: sin. Ee) :: EAe: Ee=
sin. AE X COS. AH X sin. AH
But in the triangle EнA, 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*) we have (per spherics)
sin. A E 1 (rad.): co-t. E: co-t. AH,
(sin. AH): (sin. EH)2:: (sin. E)2: 12(rad)2
Whence we get, sin. AE x Co-t. AH x (sin. AH)2=(sin. EH)2 × co-t. E X (sin. E)2. But co-t. AH X sin. AH CO-S. AH × 1 (rad.), and co-t. E X sin. Eco-S. EX 1 (rad.): therefore sin. AE X co-S. AH × sin. AH = (sin. EH)2 × co-s. E X sin. E; and, consequently, k × kx co-s. E X (sin. EH)'='
sin. AE X Co-s. AII X sin. AH
Let now the sun's longitude EH be denoted by z, considered as a flowing quantity; then, (sin. z)2 being:
EH)'=kXCO-S. EX (1 -co-s. 2 z). rotation pp, in the time that the sun's
co-s. 2 z, we shall have k X co-s. EX (sin. But the angle described about the axe of longitude is augmented by the particle Ż,
will beŻ. Therefore, by the general proposition, as 1:4 × co-s.
X E (1 co-s. 2 z) :: --×Ż: ÷ k× × co-s. E × Ż-Ż co-s. 2 z, the true regress of regress of the equinoctial point E, during that time: whose fluent, ↓ h×× co-s. Ex (z — + 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 = 4 k× X
co-s. E× an arch of 90°, the quadruple of which, or k×× co-s. E X 360° (=-3x X co-s. EX 360°) is therefore the whole annual precession of the equinox caused by the sun. This, in numbers (taking
M. Silvabelle, in his essay on this subject, inserted in vol. 48 of these Transactions, makes the quantity of the annual precession of the equinox, caused by the sun, to be only the half of what is here determined. But he appears to have
* No error arises from considering the triangles EAe and AEH, as being formed on the surface of a sphere, though the earth itself is not accurately such. The angle EAa representing the effect of the solar force, is properly referred to the surface of a sphere; therefore after its measure is truly determined, the figure Arap is itself taken as a sphere, in order to avoid the trouble of introducing a new scheme.-Orig.
+ Page 436, Vol. 10, of these Abridgements.
fallen into a twofold mistake. First, in finding the momenta of rotation of the terrestrial spheroid, and of a very slender ring at its equator: 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 thence arising, with respect to the spheriod (by reason of its near approach to a sphere) will be inconsiderable; but, in the ring, the case will be quite otherwise; its equinoctial points 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 only the half of the true value; by means of which 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
M. Cha. Walmsley, in his Essay on the Precession of the Equinox, printed in this last volume (of the Abridgement, p. 17) 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 3d 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 abstractly. No regard is therefore had to the lengths of the unequal levers, by which the particles are supposed to receive and communicate their motion: which doubtless ought to have been included in the consideration.
3 co-s. 23° 29' 1
In the 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 centre, in the time of one sidereal day. This motion he finds to be = in order to infer from this 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 centre, will be only 2-5ths of the force of an equal number of particles uniformly disposed round the whole circumference of the 3 co-s. 23° 29′ equator, in the manner of a ring. The quantity
366 X 360o
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 by which the ring tends to turn about its diameter is supposed equal to the force by which 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 hence in deriving the motion of the equinoctial points of the earth itself, he differs from him; and in the corollary to his 3d 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, though equal to that of the ring, has yet a different momentum, arising from the different situation of the particles respect to the axis of motion.
But since the aforesaid quantity, from which 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 230-229' to of 2302; whence the quantity sought comes out= × 360°=21′′ 6": which is the same that we before
3 co-s. 23° 29' 1
found it to be, and the double of what this author makes it.
What has been hitherto said, 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 density of the moon being so assumed, as to give the maxima of those inequalities, such as the observations require; in consequence of which, and from the law of the increase and decrease (which is rightly determined by theory, though the absolute quantity is not) a true solution, in every other circumstance, is obtained.