20 THE SATURNIAN SYSTEM. [§30 



When the quantity 



/2 I 2 



r' -\- zr 



is so large that a 2 can be rejected in comparison with it, the values of the functions of 

 § 28 become 



n (,') M 



V (/* + Z 2 )' 



INI 



Z (''') = t7 ^!-7 E W; 



whence, the corresponding values for r" are 



M 



n(/') = 



V( 1 +^)' 



Mr" 





h/(i+#] 



When 2 is so small that its square can be rejected in comparison with r, these values 

 become 



z(o=5, z(o=^. 



30. When the ring is very thin, and the attracted point is so near its plane that 

 the second dimensions of s and h can be neglected ; and when the ring extends inward 

 to the very axis itself, its outer radius may be denoted by a, and the points / and r" 

 may be connected by the equation 



and we may suppose 

 which gives 



(282) 



