DIRT5CT ATTRACTION OF A SPHEROID. SS75 



and g zz ~ ./, to which if we add e, we find e-\-g 



25 ji — 15 



— ^ ^ . fzz ^f; and this is the celebrated theorem 



10,1 — 6 -^ '•^' 



of' Clairant. 



It remains to he f«hown, tliat the dimintitlon of the at- Variation oif 

 tractive force at different par*:--; of the spheroid varies as.the gravity. 

 bquare of the cosine of the latitude. The elevation, being 

 every where proportional to the square of the distance from 

 the axis, may be divided into two parts; one proportional 

 to the square of the sine of the dlstaiice from the meridian 

 of the place, and the other to the distance from the pkne 

 of another meridian perpendicular to it: but the first of 

 these being constant, whatever may be the position of the 

 place to be conbi^ieied, the second only produces the varia- 

 tion. Now if we take in the second portion the mean of 

 the elevations at any two points of a less circle equidistant 

 from the meridian, it will be proportional to the sum of 

 the squares of the distance of the centre of the circle ftcm 

 the axis, and of the cosine of the distance from the meri- 

 dian in the same circle, reduced to a sitnilar direction, that 

 i§, diminished in the ratio of the radius to the sine of the 

 latitude, since twice the sura of the squares of any two 

 quantities is equal to the sum of the squares of their sum 

 and their difference. We have therefore two quantities, 

 varjnng as the square of the cosine, and as the square of 

 the sine of the latitude respectively : but the square of the 

 " sine may be represented by a constant quantity diminished 

 by the square of the cosine : and the decrease of the attrac- 

 tion of the inscribed sphere is as the elevation, which is as 

 the square of the cosine ; the centrifugal force reduced to a 

 vertical 'direction is also as the square of the cosine. We 

 have therefore, beside two constant quantities, two negative 

 forces and a positive one, all varying as the squares of the 

 cosine of the latitude ; and it is obvious, that the joint re- 

 sult of the whole, or the upper real diminution of gravity, 

 must also vary in the same proportion. 



A. B. C. D. 

 29 June, 180S. 



T 2 V, 



