71-73] Solution in Powers of a Parameter 73 



Inserting this value into equation (174), and equating coefficients of 

 successive powers of e, we obtain 



/% 



ax" 



8 ^_ iv etc 



- 22 * 



The first equation shews that g l must be a function of f , 77, f only, say P. 

 Write P f for 8P/3 etc. and put 



A = \-letc ................ ............ (176), 



a 2 .4 



so that = etc. 



8X A* 



Then the equation giving g 2 becomes 



8 #2_ !V 8A p i2 



ax~ ~ 4 * ax^' 



so that g^ 



here Q is a function of f , 77, f only. Similarly we find 



w 



where R is another function of f, 77, f only, and so on. 



This determines the value of ^ . To find u, v separately, we return 

 to equations (164) (166) of p. 71. 



Assume for u, v expansions of the form 



u = eu l + e?u 2 + e s u s + ........................... (179), 



v = ev l + e*v 2 + e?v 3 + ........................... (180). 



The coefficients in these expansions are of course not independent of 

 those in the expansion (175) already assumed for u/(l +v). The relations 

 between them are readily found to be 



................ ....(182), 



1) etc ......................... (183). 



