8 (£)] THE SATURNIAN SYSTEM. 



2^ 



! sin- (jr 



" /, [r*sin'g>+( Z -6)s] 



/' r 3 sin 2 <p (fl — Q i -\-f 2 r cos qp -f- g r cos qp) 



o 



2© 2^ 

 i' r 8 sin 2 qp cos <p (' r" sin 5 qp [r- -\- (z — b)- — prcosqo] 



~~ J<t> r'-sin , qp + (z— b) 2 J$ ~ f„ [r 2 sin 2 <p + (z — 6)-] ' 



The first term of this last memher occurs in £2'^ and £2'^ just as in £2'^ and £2'^, and 

 may, therefore, be rejected ; which leaves 



l'\ l/^-l \1 f [r 2 sin 3 qr + (z — b)-— (z — «) 2 1 [r 2 +(r — 1,)- — <, r cos gi] 



| /• cos w loo- ( /; + C> — >' cos (f ) = — / L - i— t-i ; r „\ „ — i-L L -— L=^= — ' * ^ J 



2-3 2<3 



,7.\ r ,* + {z—by—,r^^, f~_A\ a f y "+(~ — h f~ grcosqp _ 



lj X /a "^ ' Jf/« (»*+(«— 6)*— rWg,)' • 







again we find 

 /%n2ytan" ( ^ osy)(j -^ =fhcos2y ^^^ 



,/uL ' _/., r sin qp J ,/e L ' f :, i~ sm~ cp -\- (o — ruosqr)-(i — by J 



u ' 



= (0—b) f \i r COS 2 O / j Cr — P c o? g=) — (g - rogsgQr g sin 2 qpl 



1 j ./ r y /, c^ 2 sin 2 ,>+(«— A) 2 ]/;;,,) J 



The fraction, under the sign of integration, may for a moment be denoted by/^V; 

 thus 



V — i v nn« 9 rr, - ^ ^ ~ g C0S ?) ~ (g ~ r C0S f f ) r P s i " 2 9 



v — . 2 ; cos -j a) f^t 1 '* — i — / tvh>> ■> 



[r z sin 2 qp + (2— &) 2 ]/o«» 



which can be developed into the form 

 The value of M is found by putting 



/0(0) = ") 



in the value of 



* J 0(0) > 



that of N is found by putting 



r 2 sin 2 (p = — (g — b) 2 , 

 in the value of 



V[r 2 sin 2 9 ) + (g — bf]; 

 and that of P is found by division. 



First, then, to find M, the equation 



/o(0) = >" + V 2 — 2 r Q cos 9 = °> 

 gives the reduced values 



f\ = {z-bf, 



2co S a> = £+r = ^±^ 



(269) 



