AND THE FIGURE OF THE EARTH. 95 



21. The equation of equilibrium will be formed exactly as in (16.), except that 

 the expression for the velocity must be found more accurately. By (17.), the force 



of gravity at the equator is47r^^(^l+g ^J ; and the centrifugal force Isu'^a; 



therefore m = - 



4 



MXT 3^' and a;2 = 4 TT ^ 3^ (^m + W 6 - -2" j. 



The equation of equilibrium becomes 



/^^ = C = V + 2.,rM.-^^)^%(.+ .-^), 



P 



which at the surface becomes 

 C' = — 



Mtt/, , 2 I A d\ 1 /" 9 1\ /I . o 9NNa 1 / . 6 o , 3\Po 

 J.^-^%1 o 2nM«/' . 3m«\ 



Equate the coefficients of li? and those of ^/A, and we have two equations for deter- 

 mining g and A, thus showing that equilibrium is possible. These are. 

 Ma/ m 3 , 3 A Na/1 6\ 2 Pa 



3^V^-¥~Y^^ + T^V- a3-vy-yj-2i^ = ^' 



J Ma ,^ , , Na 3e , 1 Pa 



and _(A + mg)-^^ + y-^=0; 



, Ma/ m , 6A , 3 „ , me 11 o\ Na 



whence -^ (^g _-+_+_ ^2 + ____ g2j^ __, 



1 Ma/ 021^^ A 1P« 



and — (^A - 3 g2 4. -^ j = - ^ "^ • 



22. The resultant attraction in the direction of r, is obtained by differentiating 

 /— r, as found in (21.), and changing its sign. This produces 



fMa 3/„ l\Na,5/, 6„,3\Pa . „, / , ^m^\Ua^ 



4^q37^~TV^ -¥;7^+-9 1^-7^+35/7^-^(1--^ )V^ + ^^ — 2-Jy^l' 

 since the terms arising from differentiating the expression for V in (20.) with respect 

 to a, cancel each other. 



For r write its value, and arrange the result according to powers of p, and this 

 formula becomes 



y^t(l-m + mg-^' + i^2(2e + m + 2mg-^-f^)+^M2A + g2_^,^) 



47rf 



Na/ 1, 2/3 4v,, , 12 \, Pa/1 2^^ x ^ ^\ 



-^(- 5 +^ U- 5-0+^'5-V +-^V21-^'2l+^*9; 



Mar ^ , 3m 27 , 9 2 ^ ^ , 3 



= 4'^?372|l+2-^-l4^^+-4^ "7'" + T^ 



+ ^2 ( A ^ _ g + ^2 ^ g _ :^ ;„2 _ ^ £2 ^_ ^ a) + ^^ (4 g2 - 3 A - y m g) j , 



