ON THE STABILITY OF THE MOTION OF SATUBN'g RINGS. 325 



Differentiating (32) with respect to t, we get two other equations 



A^ sin a &c. = F cos/ ........................... (43), 



A^ cos a + &c. = Fsmf ........................... (44). 



Bearing in mind that B l} B^ &c. are connected with A lt A t , &c. by equa- 

 tion (33), and that B is therefore proportional to A, we may write B = Af$, 

 where 



2o>n + - 



R- * 



P- -- 



yS being thus a function of n and a known quantity. 

 The value of or then becomes at the epoch 



cr = Aj3 l sin (ww + o^+.&c. = (7 cos (ms + g), 

 from which we obtain the two equations 



sin a, + &c. = Gcos g .............................. (45), 



oj + cfec. = 6r sin<7 ........................... (46). 



Differentiating with respect to t, we get the remaining equations 



^ coso, + &c. =Hcosh ......................... (47), 



i! sin a l + &c. = H sin h ........................... (48). 



We have thus found eight equations to determine the eight quantities 

 A 1} &c. and 04, &c. To solve them, we may take the four in which A 1 coBa l , 

 &c. occur, and treat them as simple equations, so as to find ^4 j cos a,, &c. Then 

 taking those in which A 1 sin a,, &c. occur, and determining the values of those 

 quantities, we can easily deduce the value of A 1 and Oj, &c. from these. 



We now know the amplitude and phase of each of the four waves whose 

 index is m. All other systems of waves belonging to any other index must 

 be treated in the same way, and since the original disturbance, however arbitrary, 

 can be broken up into periodic functions of the form of equations (37 40), 

 our solution is perfectly general, and applicable to every possible disturbance of 

 a ring fulfilling the condition of stability (27). 



