96 MR. C. J. HARGREAVE ON THE CALCULATION OF ATTRACTIONS, 



by applying the two equations deduced in last article. The total attraction normal 

 to the surface, or the force of gravity, is found from this by dividing by the cosine of 

 the angle of the vertical, or by 1 — 2 g^ (^iJ^ — ^/u^) ; whence 

 Mar,, 3m 27 , 9 o 2 2,3 



.0/5 .72 15 „ 15 o . 12 A , ./^ „ „ . 15 M 



+ ia,2(^Ym-g + ymg-^m2-yg2 + yAj+ ii.\2 g^ _ 3 A - - m g j | . 



Let X be the sine of the real latitude ; then as jO/^ == ^2 __ 4 g ^2 — x*, we get 



Ma r 3m 27 , ^ 2 ^ , , 3 . 



^ = ^'^^3^|l + '~"2"-]4^'+"4^-7^+y^ 



+ X2(|-m-g + ymg-^m2 + ^g2 + ^A)-X4(2g2-H3A-|mg)|. 

 Let G represent the equatorial gravity ; then 



^ = G|l+x2(|^_g_^„,g^^g2 + ]^A)-X4(2g2 + 3A~|-»ig)} 



= GJl+sin2/(|m-g- j|mg + yg2-|-A) + sin2/cos2/(2g2 + 3A-|mg)j, 



which is an extension of Clairaut's theorem. 



23. In this process A indicates the amount of deviation of the required surface 

 from the surface represented by r = a (1 — g ^h^). If the equation had been taken 



a = r < 1 + e jM*2 — (^ y e2 + B ) \u^ — yu^J -|- e2 ^u;'* > , B would have been the index of 



deviation from an elliptic spheroid. 



To apply Clairaut's theorem to this surface, we have 



g = e + I e2 + B, and A = - 1' - B ; 



whence 



g = G jl + sin2/(|-m -e + yB-^me) - sin2 /cos2 / (| we - |' +3a)J, 



which is the same expression as that obtained by Mr. Airy in the Philosophical Trans- 



ft m^ f* rt ^^ (^ 



actions, 1826, except that instead of e or , the symbol is used to represent . 



24. The circumstance of the terms arising from the differentiation of V with respect 

 to a vanishing, affords an easy method of extending Clairaut's theorem indefi- 

 nitely, without calculating the value of V. 



It may be shown independently, that these terms cancel each other in all cases. 

 The (n-\- l)th term of V for the portion including the point is 



The corresponding term for the other portion is 



