METHOD OF TRANSFORMING AND RESOLVING EQUATIONS. 301 



For in all such cases the expression (33), obtained by the first 

 part of the process, becomes 



y=f{x) = (A'" + a Miv) (^ (^) + xM'vx ; . . (37.) 

 and since this gives, by the nature of the roots Xy, . . s^, 



we find, by the law (5) of the composition of the coefficients of 

 the transformed equation in y, 



C' = c(A'"+«MiV)3, (39.) 



the multiplier c being known, namely, 



c= -(p (.rj <p (x^) <p (xg) - <p (a-i) <f) (^2) <p {x^) - &c. (40.) 



and being in general different from 0, because the three first of 

 the seven terms of the expression (21) for y can only accident- 

 ally suffice to resolve the original problem ; so that when we 

 come, in the second part of the process, to satisfy the condition 



C' = 0, 



we shall, in general, be obliged to assume 



(A'" + a Miv)3 = 0, (41.) 



that is, 



A'"+aMiv=0; (42.) 



and consequently the expression (37) for t/ reduces itself ulti- 

 mately to the form which we wished to exclude, since it becomes 



y = AMivX. . (43.) 



Reciprocally, it is clear that the second part of the process, 

 or the determination of the ratio of A'" to M^^ in the expression 

 (33), cannot conduct to this useless form for y unless the two 

 functions <p (x) and p^ {x) are connected by a relation of the kind 

 (36) ; because, when we equate the expression (33) to any multi- 

 ple of X, we establish thereby a relation of that kind between 

 those two functions. We must therefore endeavour to avoid 

 those cases, and we need avoid those only, which conduct to 

 this relation (36), and we may do so in the following manner. 

 [4.J Whatever positive integer the exponent v may be, the 



power x' may always be identically equated to an expression of 

 this form, 



X' = s,^'^ + */') X + s,^'^ x^+...+ s^'^ x^- ' + L^') X, (44.) 



V 5 *i 5 «2 5 • • • * being certain functions of the expo- 



nent v, and of the coefficients A, B, C, . . . of the proposed po- 



