VOL. L.] PHILOSOPHICAL TRANSACTIONS. 175 



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 dia- 

 meter 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 

 in respect to the axis of motion. 



But since the aforesaid quantity, from which the motion of the earth's equi- 

 nox 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 a of 230^; whence the quantity sought comes out = 



3co-3.23"29'_ 1 w23i»^-«229^ ^^^o nMif\''' U' 1 ' *u ^u ^ u r 



V^ad ^ 366^ ^ — 23b^~" ^ =2r' : which IS the same that we before 



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. 



