§30] THE SATURNIAN SYSTEM. 21 



and from the formulae of §§ 12, 20, and 26, 



11 ('"') = Wa C0Sec2 * i{ V ] - C0S/F? )' 



i2(r") = ^sec a n'(I?+co S zFJ) J 



R ( / ) == H sec2 ^t(l-^m 2 e)FJ-Ea 



'©a 2 



■?- E (/) = - 



a r 



2 Ma 



Z (/) = ££= sin 2 H (sec * I? — FJ) = ^f J2 (r'), 



r*, „. 2M 2Mz 2l ., .„, . „,. 2M 2 cos 4 4 1 r> , „\ 



Z(r ) = -i 7=^-3 cos 2 ^(sec?EJ4"FJ) = ^ 5 — h ■"(>*)> 



v y a 2 'Q a 3 v lit/ a 2 a 2 cos 2 j v ;> 



a E (/') + 12 (/) = ±| sin 2 1 ,=£[« R (/) + 12 (/')], 



i2(0 = ^2(/) + ^(/-r")R(/). 



If /■' is so large that the square of the lineal dimensions of the ring can be rejected, 

 these quantities become 



n (/) = f, n (/o = ?* 



B(0 = 5. R(/')=^ = ^, 



v y /- ^ ' of a 2 ' 



z(/)= m ; Z (o=^-^. 



v y r ' 3 v ' a- a 3 



The general values of this section can easily be reduced to series. If 



b b 



IT; tfj i and S t y i 



a a 



denote respectively the continued product of all values of the function yi i from a to b 

 and the sum of all these values, we find 



n ,,, M . 2M-r</2i— 1\« 1 /a\ a, 'l 



(283) 



