218 THEORY OF NEWTONIAN FORCES. [PT. I. CH. V. 



(4) "TF^PJo W^7 



:>^ = ]*~ l 4/iq^J - 1 (** + ry} , 



F-9 ^ f_ 2(6 2 -a 2 )g 2 5 3 c?5 



4 - 2woWa?J - ^TI^(^I^ 



^ 



/" s 2 c?5 s 



] (i=j- VfTJ- 



ds 



so that 



(7) z =(S5{v / fe?- sin " 1 V // 



(8) F = 



/6^-a 2 



V 62 ^ 



Now 

 so that 



r S 2^ s _ 



I = J (sin" 1 s s v 1 s 2 }, 



, N ^ 27ra6 2 y f . 6 2 - a 2 (6 2 - a 2 )(a 2 + o-)) 



(9) F =(6 2 ^)i{ sm - 1 VFT,-V- -FT^ 



TP . /6 2 -a 2 ., /6 2 - a 2 



For sin" 1 A / - 7 - we may write tan" 1 A / - , 

 V o 2 + <r V a^ + <r 



, .- 6 2 - a 2 



^ 



- 2 



then 



These formulae all serve for an oblate spheroid, where a < b. 

 For a prolate spheroid, b > a, they introduce imaginaries, from 

 which they may be cleared as follows. 



