109-1 n] Rotating Cylinders 109 



This is to be a solution of equation (310), so that the following equation 

 must be an identity : 



psfr +..,) 



(319). 



To avoid the useless printing of terms which would ultimately be found 

 to vanish, we shall at once strike out all coefficients s C n of which the true 

 value is zero. Accordingly, in place of the general equation (317), we assume 

 separate equations of the form 



f j = 3c 3 77 2 + G! - c_!?7- 2 - 3c_ 3 77~ 4 - 5c_ 5 ?7- 6 - 7c_ 7 77~ 8 (320), 



f * - 6/ 6 r7 5 + 4/ 4 77 3 + 2/ 2 77 -f ./o^r 1 - 2/_ 2 77- 3 - 4/_ 4 77- 5 (323), 



and so on. In these equations terms such as . d^" 1 have no value but are 

 written in for completeness. The quantities d , f etc. do not themselves 

 vanish but represent the quantities 2 (7 , 4 (7 etc. which may have finite values. 

 Each of the series f 1} f a > fs, &, extend to infinity, but we shall not require 

 more than the six first terms written down to give the approximation to 

 which we are working. We shall assume three similar series for f 5 , f e and f 7 , 

 the coefficients being denoted by the letters g, h, i respectively. 



111. Since equation (319) is to be an identity for all values of 6 and 77, 

 we may equate the coefficients of and shall obtain a system of equations 

 which must be true for all values of 77. The equations obtained by equating 

 coefficients of 6, 6, $ 2 , 6 s , ... are found to be as follows : 



(324), 

 .' ........ (325>, 



2 ) + d 4 (^ + f <). . .(326), 



4 ) 



+/o +/ 2 (^? 2 + f o 2 ) +/4 (^ 4 + ?o 4 ) +/ (^ + ?o 6 ) ............... (328), 



and similar equations. 



