ai4 



PHILOSOPHICAL TRANSACTIONS. 



[anno 1738i 



10 — 2pcfxe 



1+^ 



+ 



10 — iqcg>e 



+ ? 



'■=^^=:r. Bat it is easy to perceive, that x is pro- 



3 + PX5+P Z-\- qy. S + q 



portional to the square of the sine of the arc pm, or of the complement of the 

 latitude. Whence we may therefore conclude, that the diminution of the 

 gravity from the pole to the equator, is proportional to the square of the cosine 

 of the latitude ; or, which is the same thing, that the augmentation of gravity 

 from the equator to the pole, is as the square of the sine of the latitude, as Sir 

 Isaac Newton has demonstrated in his hypothesis of a homogeneous spheroid. 



]8. From the following calculation it is easy to conclude, that Sir Isaac's 

 theorem, (Prin. Math. lib. 3, prop. 20) which is this, that the gravity in any 

 place within, is reciprocally as the distance from the centre, cannot obtain here. 

 For we may see by the foregoing expression, that the gravity in n cannot be to 

 the gravity in p, as 1 to 1 + ^> except when p = g = O, which happens only 

 in Sir Isaac's homogeneous spheroid. 



It was for want of considering, that this theorem was demonstrated by Sir 

 Isaac only in the case of his homogeneous spheroid, that several geometricians 

 have too hastily concluded, that this theorem might be applied to determine the 

 ratio of the earth's axes, and the lengths of the pendulum observed in two places 

 of different latitudes. Dr. Gregory is one of those who have fallen into this 

 mistake, in his Elements of Astronomy, lib. 3, sect. 8, prop. 52. And in the 

 Philos. Trans. N° 432, it is concluded, from the proportion of gravity at 

 Jamaica to that at London, that the diameter of the equator must exceed the 

 earth's axis by the JQOth part, which computation was founded on this 20th 

 proposition, lib. 3, of Sir Isaac's Principia, which is true only of his spheroid. 

 The Manner of priding the ^xes of the Spheroid, the Variation of the Densities 

 of the Strata being taken at pleasure. 



IQ. Let us now suppose, that the centrifugal force at the equator is known 

 by observation, as also within the earth, &c. and that it is a certain part, as the 

 mth part of the gravity ; by articles 14 and l6, we shall have this equation : 



2c/i 



■ +;■ 



+ 



2p- 



2c/e'+^« ^cge'-^f '2q - Zcge^ + ?» 8cfme' + ?a 



+ 



+ 



Scmge^ "' ^« 

 "*" F+qxT+q' 



S+p ' 3 + px5+p ■ 3 + q 3 + qx5 + g 3+px5 + p 



From hence it will be easy to derive the value of «, because y, g,p, q, will be 

 given, from the hypothesis that will be chosen, for the variation of the density 

 in the internal parts of the spheroid. 



20. And if on the contrary a. be given, that is, if we know by observation 

 the ratio of the axes of the planet concerned ; then by the foregoing equation 

 we may perceive, whether we have assumed an agreeable hypothesis for the 

 variation of the densities : but we cannot precisely determine what this hypo- 



