ON THE STABILITY OF THE MOTION OF SATURN'S KINGS. 347 



2nd. Variation due to disturbance of first ring. 



If we put a(l+p) for a in the last expression, we get the attraction 

 when the first ring is displaced. The part depending on p is 



-- , - ?r s P inwards ........................ (98). 



ira (a ay 1 



This is the only variation of force arising from the displacement of the 

 first ring. It affects the value of L in the equations of motion. 



3rd. Variation due to waves in the second ring. 



On account of the waves, the second ring varies in distance from the 

 first, and also in mass of unit of length, and each of these alterations produces 

 variations both in the radial and tangential force, so that there are four things 

 to be calculated : 



1st. Radial force due to radial displacement. 



2nd. Radial force due to tangential displacement. 



3rd. Tangential force due to radial displacement. 



4th. Tangential force due to tangential displacement. 



1st. Put a'(l+p') for a', and we get the term in p' 



-7 



2nd. By the tangential displacement of the second ring the section is 

 iced hi the proportion 

 of the radial force equal to 



reduced in the proportion of 1 to 1 -, , , and therefore there is an alteration 



inwards = /A' -1-7 say (100). 



ira' (a -a') ds' ' ds' 



3rd. By the radial displacement of the second ring the direction of the 

 filament near the part in question is altered, so that the attraction is no longer 

 radial but forwards, and the tangential part of the force is 



''&' forwards ................. (101). 



x 



TTO, (a a) ds ds 



442 



