[ 193 ] 



XXVII. Note on some Objections of Mr. Cayley and Mr. Cockle. 



By G. B. Jekrard*. 



HAVING been unwell for some weeks past, I have had no 

 opportunity of expressing what I thought of the objec- 

 tions referred to at the close of my last paper f. In appearance, 

 these objections are most formidable. They were intended to 

 upset a conclusion of mine respecting the existence of an excep- 

 tional case, hitherto unobserved by mathematicians, in Lagrange's 

 theory of homogeneous functions. My conclusion, however, still 

 seems to be unshaken. But there is assuredly a notable error 

 somewhere. Let us try to find out with whom it rests. 



1. The objection which comes first in order is so peculiar, 

 that I am quite unable to characterize it beforehand, or to reduce 

 it to any known class. 



"Putting, for shortness," says Mr, Cayley, 



««=/(<)/(. W)/(' 4 ), 



I understand Mr. Jerrard to mean that a, a 5 being given as the 

 roots of two equations of the sixth order respectively, ct cannot 

 be expressed as a rational function of a 5 and of the coefficients 

 of the two equations ; or, what is the same thing, since the co- 

 efficients of each equation are rational functions of the coeffi- 

 cients of the given equation of the fifth order, that a cannot 

 be expressed as a rational function of a. 5 and of the coefficients 

 of the equation of the fifth order. If this is not Mr. Jerrard's 

 meaning, he will doubtless set me right ; but that u can be so 

 expressed seems so clear, that I must apologize for giving a 

 formal proof of it %." 



The question, however, relating to a. is, as we shall see a little 

 further on, very^different from what it is here implicitly assumed 

 to be. 



2. Let us suppose, in conformity with my previous notation 

 for n = 6, m = 5§, that u and v designate two rational six-valued 

 homogeneous functions of the roots of the equation 



# 5 + A 1 a? 4 + A 2 a; 3 + ..+A 5 =0; 



then by Lagrange's theory we can generally express either of 

 these functions in rational terms of the other and of the coeffi- 

 cients Aj, A 2 , . . A 5 . The only exception, according to that theory, 

 is when there are equal roots in (U) or (V), the equations on 

 which u, v respectively depend, — or rather when the number of 

 unequal roots in (U) differs from that of unequal roots in (V). 



3. But I maintain that when u and v are taken equal to those 



* Communicated by the Author. 



t Philosophical Magazine for June 1862. X Ibid, for March 1862. 



§ Ibid, for May 1861. 



