51'2 VHIOSOPHILCAL TRANSACTIONS. [aNNO 1/32. 



And in the case of the gravity being in tlie simple reciprocal ratio of the 

 distance, it will be log. - = — ■^, or the log. of ca — log. of cp = -^. 



It appears that, 71 being an affirmative number, either integer or fraction, 

 that is in every hypothesis of gravity being directly proportional to any power 

 of the distance, the equatorial diameter will always be greater than the axis of 

 rotation. But \t' n be any negative number, that is, if gravity be inversely pro- 



I 

 portional lo any power of the distance, then will ca : cp :: (2p)'-" : (2p — nf 



— J)'""- Now if n be less than 1, make k = 1 — n, and there will be 



CA : CP :: (2/j) -^ : (2/j — k/fi ; but if n be greater than 1, make ?i — I = k, 



' I I I 



and then ca : cp :: {2p) - Z' : (2/j + k)~, or ca : cp :: (2p + 4^) * : (2/?) *". Fur- 

 ther, when n = — 1, we found that log. of ca — log. of cp = ■^, So that 

 it appears there is no hypothesis in which the equatorial diameter is not greater 

 than the diameter of the meridian. 



It is sufficiently apparent that the figure of the spheroid depends on the ratio 

 of the centrifugal force to gravity. Now what that ratio can be may always 

 be seen in any hypothesis, and that hence the figure of the spheroid will 

 result. 



If gravity be supposed uniform, it will be n = 0, and then ca : cp :: 2p : 

 2/j — f. Therefore in the earth, where the centrifugal force at the equator is 

 equal to the 289th part of gravity, if there be sought the ratio of the equatorial 

 diameter to the axis in the hypothesis of a uniform gravity, putting 28Q for p, 

 and 1 for/, then ca : cp :: 578 : 577. 



The centrifugal force may be equal to gravity, which would happen if the 

 diurnal rotation were 17 times quicker; and then ca : cp :: 2:1. But should 

 the rotation become greater and greater, the parts would be successively dis- 

 sipated, till at length the earth would be reduced to a mere atom. Hence it 

 appears that in this hypothesis of a uniform gravity, the earth cannot be more 

 depressed at the poles than when the equatorial diameter is double the axis of 

 rotation. In this case the earth will consist of two paraboloids, like as found 

 by Huygens, in his tract on the Cause of Gravity, for that particular hypo- 

 thesis which he only examined. 



If gravity be made proportional to the distance from the centre ; then will 

 « = ], and CA : CP :: -//j : \/ {p — f). If therefore the centrifugal force be 

 equal to the force of gravity, the equatorial diameter would be infinitely greater 

 than the axis of rotation : that is, the spheroid would become only a circular 



