404 Prof. Collins on the Attraction of Ellipsoids 



and .'. the differential of the required attraction on C is 



47rabc cos^ 6 sin d .dO 

 (c^ sin2 e + a- cos^ e)^{c^ sin^ d + b^ cos^ 6)^ ' 



by putting cos^=w, or tan^^=t;, this expression, given by 

 Poisson {Mecanique, vol. i. p. 190), becomes transfonued into 



47rabcu^du 



or 



2'jrc . dv 



(1 +«)^(l + .)(l + ^.)(l + g.) 



5. Now supposing a > i > c, let OA' and OB' portions of OA 

 and OB be the semiaxes of the focal ellipse whose plane is perpen- 

 dicular to OC, then OA'2=«2_p2 ^nd OW^=b''-c-; let OQbe 

 perpendicular to the tangent PTQ, which touches this ellipse in 

 T and meets OA' at the point P ; let < OPT = ^, and 



p2 = c2 + (i^ - c'^)u^ = c2 + OB'2 cos^ e 

 and 



p'2=c2 + (a2_c^)„2 = c'^_|. OA'2 cos2 6'; 



OP p OA' 



and if the point P be taken such that pri-, = — or OP = p, 



'^ OA' c c '^ 



then the equation 



OA'2cos2A'OQ+OB'2cos2B'OQ=OQ2; 

 .'. = OP^ sin^ ^ gives 



ar—c 



{a^-c^) sin2./) + (62_c2)cos2<^=— -2^(c2 + J2_c2.cos2^) sin2</>, 

 which gives 



{a^-c^) cos2 ^=cos2 (/>(c2 + «2_c2 . cos2 6), 



OA' 

 i. e. OA'^cos^ ^=p* cos^ 0, and /. cos 0= — j- cos 6} and since 



P 



0P=^^ p = ^^ (c^ + OB'2 cos2 d)^, 



.-. Pp = rf . OP = — (c^ + OB'2 cos^ (9) "* X OB'2 cos 6 sin 6* . t?6' 



0A'.0B'2 a ■ a ja 



= cos dsxvLO , do. 



cp 



