AND THE FIGURE OP THE EARTH. 93 



2nd. Let w = 2, and we get 



which gives 



+ ^j ~^da! I — 4 TT f( jU»2 — yj r^ (same function of a, e^ and AJ. 

 3rd. Let w = 3, and we have 



or 



— (jjA — Yf^^ ~\~ 35 ) 4 TT ^ H (same function of a, gj and Aj). 



19. To the sum of these must be added V for the inner spheroid, for which we shall 

 have to find the value of V generally, when the attracted point is without the body. 

 The general term of the series for V is 



and 



y^r"'<pr'<^r' =7^"«'" .(pa! .da' + a^+^<pa(^^ efj.^ + /^^ (g2 - A) + f^*^^^) 



and similar equations hold for the other functions. 

 1st. Let » = 2, then we have 



^yi'i d(Ji.' j J^^\'^ (pa' da' + a^<pa(- s f^'^ + /t^'* (2g2 _ A)) + aV« 15 



^y^^a'^Fa' da' + a^Fa{- sf^'^) -\-J^\'^na' da'l , 

 which is equal to 

 l^|y^'a'2^a'rfa' + fl3^«(-y + ]-(262-A))4-a*?''«^+ jX"^''^«'-^«' 



- ^a^Fa.B + ^J^^\'^Ua'da'\. 



