174 PHILOSOPHICAL TRANSACTIONS. [aNNO 1757. 



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 equa- 

 tor. The difference indeed thence arising, with respect to the spheriod (by rea- 

 son 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 re- 

 gard 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. 1^ ; 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 equi- 

 noctial 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. 



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 mis- 

 takes 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 re- 

 ceive and communicate their motion : which doubtless ought to have been 

 included in the consideration. 



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 



J CO- s 23° 2Q' 1 



one sidereal day. This motion he finds to be = — -— ; — - x zn^, X 360'' Then 



4 raa. 300-J 



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 de- 

 monstrated 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 



r • mi • 3 COS. 2 J° -9' 2 I an 



equator, m the manner of a nng. 1 he quantity — - -^ ~ X -r X^^j^j X 300" 



