considered geometrically. 405 



Let PP' be perpendicixlav to the consecutive tangent from p, 



then p?'='Pp cos ^ ; 



cpp' 



cos2 e sin e.d9: 



but the attraction of the slice of the ellipsoid between D and E 

 on C along CO was found above 



_ 47rabc cos^ ^ sin ^ . dO 



And, moreover, as P'^ is obviously =d.WTV, 

 .-. =^(B'TP- const B'TA'), 

 i. e. =«?(TP— arc TA') ; .*. the attraction of the slice on C is 

 4:7rabc'^ ^, 47rabc'^ 



xpV'= 



d.iTF-avcTA')- 



and /. the attraction of the whole ellipsoid on C is 

 = (a-2_c2)(^2_g2j X (TyP^-arc T^A'), 



where P^ and T, denote the ultimate positions of P and T corre- 

 sponding to ^ = 0; and since by constniction 



OA' DA' 1 



OP = ^^ p = ^^^ (C^ + 62 _ c2 . COS^ m^, 



c c ' 



.'. when 0=0, OP; = - OA', and 



* * ' ' be aba 



6. Hence, also, the differential of the ellipsoid's attraction on B, 

 i. e. the attraction on B along BO of a slice of the ellipsoid com- 

 prised between two cones of revolution whose vertex is B and 

 axis BO, and semiangles are 6 and 6 + dd, is 

 47rabc cos^ dsinO . dd 

 ^ {^iQ^Y+c'^^ife} ^ {b^ sin^ + fl2 COS* 6)^ 

 _ 47rabcu^ . du 



