§10] THE SATURNIAN SYSTEM. 13 



Its derivative with reference to z is 



j __ _ r v F i 



10. When the attracted point is near either of the curved edges of the ring, the 

 value of LI'" admits of peculiar transformations and reductions. Thus let the approxi- 

 mated edge be that for which 



z = 6, o = a 2 , 

 and put 



4 = the distance of the attracted point from the 

 edge, and let this distance be so small that its cube can be rejected. We have then 



./I = (« 2 _,f+(^-£) 2 , 

 sec% = — f - = _ (1 + _) = _(1 + L + ^, 



!s -ll 1+ "IT" H 8^~ -) — T V + ~¥7" + -sis - ;' 



.__«/■. «. — r (---&) 2 -3(a 2 — r) 2 \ 



22 — 4r 2 \ 1 r _ ~ I? /' 



sec 1 



cos 2 1 



I, /-. a 2 — r /.? — 3(a, — r)-\ 



008^=^(1--^-— J ') , 



sin ? ;22 = !-i^ ^-, cos rfa = ^— — '—, 



'2 <2 



sin o r ' -_ 2v/(« 2 -^v/(z- by- 



bill _/ /yoo j7, j 



'2 



sin rfe cos 42 = sin i]^ = ^ 2 ~ = rj£', 



sin 2 ^=^(l + ^), 



cos 2 ri" -- (a *- ry (\ 4- ( g - 5 )M 

 cos »?* — — ^— ^1 + r{ai _ r) ), 



cos^=^(l_^), 



= Sin 2 t.V, ( 1 -4- -=-: 7 r^-, —- ), 



(275) 



