METHOD OF TRANSFORMING AND RESOLVING EaUATIONS. 309 



necessary to resolve any equation higher than the third degree. 

 The first part of the process is now completed, that is, the two 

 conditions (8) and (14), 



A' = 0, C = 0, 



are now both satisfied by an expression of the form 



y=f{,v) = A"'<p{x)+^''xi^)> . • • . (89.) 

 which is analogous to (33), and in which the functions <p {x) 

 and X {^) are known, but the multipliers A'" and N"^ are arbi- 

 trary; and the second and only remaining part of the process 

 consists in determining the remaining ratio (82), of A'" to N^, 

 by resolving an equation of the fourth degree, so as to satisfy the 

 remaining condition, 



D'-aB'2 = 0. . . (19.) 



[70 Such, then, (the notation excepted,) is Mr. Jerrard's ge- 

 neral process for reducing the equation of the m^^ degree, 



X = x"' + AoT-^ + B x'"-^ + C oT-^ + D ^"'"^ + E oT'^ 



+ &c. = 0, . . (1.) 

 to the form 



Y = y'" + B'j/™-^ + aB'^y"'-* + E'y"*-^ + &c. = 0, (20.) 



without resolving any auxiliary equation of a higher degree than 

 the fourth. But, on considering this remarkable process with 

 attention, we perceive that if we would avoid its becoming illu- 

 sory, by conducting to an expression for y which is a multiple 

 of the proposed polynome X, we must, in general, (for reasons 

 analogous to those already explained in discussing the transfor- 

 mation (12)j) exclude all those cases in which the functions 

 <f) {x) and X i^)) ^^ the expression (89), are connected by a re- 

 lation of the form 



X(^) =«<;>(^) + aX; . . (36.) 



because, in all the cases in which such a relation exists, the first 

 part of the process conducts to an expression of the form 



y = (A'" -I- a NV) (|> (a?) -h X N^X, . . . . (90.) 



and then the second part of the same process gives in general 



(A'" + a NY = 0, (91.) 



that is 



A'" + a N^ = 0, (92.) 



and ultimately 



y = A N^ X (93.) 



