METHOD OF TRANSFORMING AND RESOLVING EQUATIONS. 317 



greater than four, that is, ivhen the proposed equation is not 

 higher than the fifth degree. But the foregoing discussion 

 (though the great generality of the question has caused it to be 

 rather long) has the advantage of extending even to equations 

 of the sixth degree, and of showing that even for such equations 

 the method generally fails, in such a manner that it will not in 

 general reduce the equation 



a?« + Aa?^ + B^'^ + C^+Da?^+Ea:+F = (140.) 

 to the form 



/ + B'7/4 + aB'^?/^+ E'y + F' = 0, . . (141.) 

 except by the assumption 



^ = L(^« + A^ + Ba;-^ + Cr" + D ^-^ + E.r+ F); (142.) 

 which gives, indeed, a very simple result, namely, 



y = 0, (143.) 



but does not at all assist us to resolve the proposed equation 

 (140.). However, this discussion may be regarded as confirming 

 the adequacy of the method to transform the general equation 

 of the seventh degree, 



a:7 + Aa;«+ Bjr^ + Ca:4 + Dj?^ + Ex2 + Fa; + G = 0, (144.) 

 to another of the form 



y + BUf + a B'V + E'/ + F'3/ + G' = 0, . . (145.) 

 without assuming y — any multiple of the proposed evanescent 

 polynome x^ + Ax^ + &c. ; and to effect the analogous trans- 

 formation (20), for equations of all higher degrees ; a curious 

 and unexpected discovery, for which algebra is indebted to Mr. 

 Jerrard. 



[9.] The result obtained by the foregoing discussion may seem, 

 so far as it respects equations of the sixth degree, to be of very 

 little importance; because the equation (141), to which it has 

 been shown that the method fails to reduce the general equation 

 (140), is not itself, in general, of any known solvible form, what- 

 ever value may be chosen for the arbitrary multiplier u. But it 

 must be observed that if the method had in fact been adequate 

 to effect that general transformation of the equation of the sixth 

 degree, without resolving any auxiliary equation of a higher de- 

 gree than the fourth, then it would also have been adequate to 

 reduce the same general equation (140) of the sixth degree to 

 this other form, which is obviously and easily solvible, 



y + B'y + D'y2 + Y' = 0, . . . . (146.) 

 by first assigning an expression of the form 



2,=f{x) = A'" c^ {x) + W X (■*•), . • (89.) 



