300 ON THE STABILITY OF THE MOTION OP SATURN'S RINGS. 



where the values of the differential coefficients on the right-hand side of the 

 equations are those in which r, stands for r, and fa for fa 



and taking account of equations (9) and (10), we may write these equations, 



Substituting these values in equations (6), (7), (8), and retaining all small 

 quantities of the first order while omitting their powers and products, we have 

 the following system of linear equations in r lt lf and fa, 



=0 ......... (11), 



= ......... (12), 



=0 ......... (13). 



PROB. III. To reduce the three simultaneous equations of motion to the 

 form of a single linear equation. 



Let us write n instead of the symbol -j- , then arranging the equations in 

 terms of r lt 6 lt and fa, they may be written: 



=0 ...... (14), 



= ...... (15), 



- (SM) r, + (/BfcW) 0, + (BJfn 9 -SN}fa =0 ...... (16). 



Here we have three equations to determine three quantities r,, 0,, fa; but 

 it is evident that only a relation can be determined between them, and that 

 in the process for finding their absolute values, the three quantities will vanish 

 together, and leave the following relation among the coefficients, 



