14 THE SATURNIAN SYSTEM. Q§ 1 1 



1 ^ — ti" — r/ — ( " 2 ~ r) ^ z ~ h)2 



¥} a = (1 + i cos 2 %) log (4 sec 4 2 ) — £ cos 2 4a 

 E* = 1 -f- 1 cos 2 ? - 22 log (4 sec i<&) — } cos 2 %> 

 F co , s ?/ 22 = 7j^ (1 -|- 1 cos 2 %) — | sin 2 ?j^ cos 2 i^, 



Fe„, 2 *J22 = Vzi (1 4* ^ C0S2 *a) _ i Sil1 2 ^22 COS 2 ?^, 



E co , a -»/ 22 = r /22 (1 — 1 cos 2 ^,) + l sin 2 ^ 22 cos 2 4,, 

 E c0 ia rfe = rj'^(l — i cos 2 4s) + i sin 2 fa cos 2 4>, 

 F^rj^E^ *&' = <, 



H, 2 ^ = (F co , a #, — E co , : ^) F, 1 ^ — F^^ E\, + I © 



= i '0 — i;^ ( 1 ~h *■ cos 2 2 - 22) H - 1 s ' n 2 ^22 cos 2 ^ — $ cos 2 4j sin 2 1] 22 log (4 sec t^), 



H, i^ = i © — ? /22 ( 1 + < cos2 *&) — |— ^ sin 2 r^ cos 2 i& — i cos 2 4a sin 2 ^ log (4 sec 4a), 



TT " f 1 /5T\ « '" 



3.^22 = i'e) — J i22 , 



= ^ -log (4 sec i^) — | ~? — j — rw 



2010 — 5), ,. . . 2o|(z — 5) , 2o| v^« 2 — r ) 2 / l ^ '\ 



= _ ( ,_ i)r (l + 3 J^)lo g (4sec^)-r(,-i) + ^^ 6 ^|©-» /2 ). 



These quantities, substituted in the value of SI"', reduce it to 



!2£' = 2 (r — « 2 ) (2 — 5) log (4 sec 4,) + 3 (a, — r) (* — b) 

 — (* — *)» tan" 5=J - (a* - r) 2 tan" J=^. 



11. When the cube of b can be rejected, the integral of the value of SI" in § 6 

 gives by the formulae of § 8, 



2© 2<3 



i2i" = -flr=r(*/ )+jrprcoB 9 )log(/ + ? — rcosg>)] 



U 



2^ 2-3 



(276) 



7 C q" — Qr cos cp . , , r r 2 -|-z 8 — prcosqp 



_ J? /. "•' J* (** + ^ ^ »)/. 







= 25^ ( V y°°~ g)3 H, tf + H, fg') 



+ 2 b y/ [(, + ,) 2 +^] e^+ JcfSrS] pl * 



