6 THE SATURNIAN SYSTEM. [§ 8 («) 



we have in all the forms in which the function to be integrated has the same value for 

 negative as for positive values of ip', 



2-a j-a j© 



r = 2 r = 4 r. 



o -*© o 



We also have 



2© 2© 2© 



*) 



2© 



p r sin 2 qp cos qi (/.-, — z -j- />) 



X 



M/;-(^-*) 2 ] 



2-3 



f' Qr sin 2 qp cos qi (^ — a -(- 6) 



2-3 • 2© 



r ()>' sin 2 y cos (ji [ ( J\ TV 1 " s * n ~ ' cos f 



J<P /<>«» ^ V0 / 2 /o(0) 







The first term of this last member, occurring in both Sl'i and Si", may be omitted, 

 which leaves, for the present casej 



2© 2© 



(z — J) q r sin 2 qp cos qp 



/Joo^ylog (/,+•-*)]=/, 



„ /2/oco) 



u 



But 



q r sin 2 y cos cp = — or cos 8 y -|- o r cos 9 



/, 2 1 C 2 + '-' 2 , (f — r 2 )-\ r2 (Q- + r") {if — r 2 ) 2 



= (4 cos- y + ^- cos 9 + ^-/ j/ 2 

 which gives 



e r ' ■ 8fr /- " (0) 8(. 2 r 2 ' 



2© 2© 







_ ( g -Z.)( e 2 +r 2 )( e 2 -r 2 ) 2 /? 1 



Again wc find 



2-3 2© 



/ r , ,, 1 N 1 r r sin qi (D^ /:,-{-?■ sin qp) 







2© 



r- sin 2 qp (/„ -f- q) 



- */s (/*+«— rcosqp) 



J* 



(268) 



2© 



r 2 sin 2 qp (/ 2 -f- p) (^ — p -f- ?" cos qp) 



ftlf* —ie—r cos qp) 2 ] 

 



