372 Mr. J. Cockle on Transcendental Roots. 



P, and W are cyclical. Consequently he would obtain the general 

 solution of a quintic in a form which involved no quintic radicals. 

 Such a result is of course inadmissible*. 



Without reiterating objections which I urged in the last May 

 Number, let me observe that of the two systems (<r,) and (<r 2 ) 

 suggested by Mr. Jerrard (S. 4. vol. iii. p. 114), the last is in- 

 admissible. To admit it, would be to ignore the principle (a 

 neglect of which led me into an objection to Wantzel's argument) 

 that interchanges of the constituent symbols do not affect the 

 form of the function. There is no other conclusion than that 



I r«(a?«)(«i8) = 1 r«(a? / j), .. 1 »'«W(«f) : =V fl W, 



so that (o-j) must hold universally. If (<r 2 ) also holds, (o" ; ) (lb. 

 p. 115) takes the form 



(V» + r,(0)V* + r#)V + r a (0)> 1 =0, 



which is inconsistent with the supposition (Ibid. p. 112) that 

 V 15 -|-C 1 V ,4 + . . =0 cannot be depressed. Hence (compare lb. 

 p. 116) 



V* + B^ 2 + r 2 {w) V + r d (*) = 



is the form of one of Mr. Jerrard's cubics ; and in order to con- 

 struct it, we have to encounter the difficulty of solving the 

 general equation of the fifth degree, for in no other way can r{oc) 

 be determined. Apart, then, from Cauchy's theorem and M. 

 Hermite's argument, Mr. Jerrard's process presents intrinsic 

 objections fatal to its success. Let me add that although, by a 

 theorem of Abel, every value of ^H at p. 79 of Mr. Jerrard's 

 ' Essay ' be a root of (ac), so that we may deduce 



S-V{P/(^)}=0, 



degree in t, the function 



(in which h, t 3 , . . fo are the \i values of t formed by permuting all the 

 roots excepting x ) cannot be symmetric in x for all values of m, otherwise 

 the reduite would not be irreducible. Hence m may be so assigned as to 

 render 



(/)(n*)=a-f&# + . . +ex A ; 



and x Q , which may be found as a rational function of 0, can contain no other 

 radicals than those which enter into t ly t 2 , . . t^. 



* The moment we proceed to the practical application of our formula;, 

 we are led to conclude that in those cases in which the root is expressible 

 by quintic (with or without quadratic) surds, the sextic in 6 has a rational 

 linear factor ; and that when cubic radicals appear, the given quintic and 

 its resolvent sextic are, each, reducible. i 



