300 SIXTH REPORT — 1836. 



MlvJ (^2.) 



which remaining ratio can then in general be chosen so as to 

 satisfy the remaining condition 

 C' = 0, 

 without our being obliged, in any part of the process, to resolve 

 any equation higher than the third degree. And such, in sub- 

 stance, is Mr. Jerrard's general process for taking away the 

 second, third, and fourth terms at once from the equation of the 

 wi"* degree, although he has expressed it in his published Re- 

 searches by means of a new and elegant notation of symmetric 

 fimctio7is, which it has not seemed necessary here to introduce, 

 because the argument itself can be sufficiently understood with- 

 out it. 



[3.] On considei'ing this process with attention, we perceive 

 that it consists essentially of two principal parts, the one con- 

 ducting to an expression of the form 



3,=/(^) = A'" 4i(^) + M^^X (•*•), . . . (33.) 

 which satisfies the two conditions 



A = 0, B' = 0, 

 the functions (^ (.r) and ^ (^) being determined, namely, 



A' X' A" X" , X'" ... . 



^ W =-i^' ^ "•" A'" '^ ■•" "^ ' ' • • ^^ ^^ 

 and 



M' ^' M" u" M" f^>" ^iv 

 X \^i — j^Jrv •*■ + jjjrv ^ + -^iv "^ + * j • • \9^') 



but the multipliers A'" and M'^ being arbitrary, and the other 

 part of the process determining afterwards the ratio of those two 

 multipliers so as to satisfy the remaining condition 



C' = 0. 



And hence it is easy to see that if we would exclude those use- 

 less cases in which the ultimate expression for the new variable 

 y, or for the function fix), is a multiple of the proposed eva- 

 nescent polynome X of the m"* degree in x, we must, in general, 

 exclude the cases in which the two functions cji (x) and ;^ (jt), 

 determined in the first part of the process, are connected by a 

 relation of the form 



X(.r) = «(f»(^) + XX, (36.) 



a being any constant multiplier, and A X any multiple of X. 



