o/Mr. Cayley and Mr. Cockle. 195 



We thus perceive that we cannot derive any aid from Lagrange's 

 theory in expressing a 5 as a rational function of a and of A p 

 A& . . A s . It remains to show that (e 2 ) also is inapplicable when 

 u = &, v = a b . 



The equation (e 2 ) is supposed to be obtained directly from the 

 theory of Lagrange. But an equation identical in its ultimate 

 form with (e 2 ) ought to be found by combining (e^ and (U). 

 Whence 



u=q-, ... {e l3 vy 



if 



since (ej here becomes, or at least ought to become, a 5 =a 5 . 



Of the interpretation of (e v U)', which is indicative of an in- 

 congruity, I intend to speak in another place. 



6. The curious irrelevancy of Mr. Cayley's objection will now 

 be seen. He merely occupies himself in proving what I had 

 taken for granted, while he leaves untouched the main question 

 to be discussed, which is this : — Is Lagrange's theory inapplicable 

 to the case ofu = ct, v=a 5 , in virtue of the equation (e\) ? 



7. Mr. Cockle, indeed, notices this equation in a separate 

 paper in the same Number of the Philosophical Magazine as 

 that which contains Mr. Cayley's objection. But the meaning 

 of (e^) is so misapprehended by Mr. Cockle, that the very exist- 

 ence of the equation might just as well have been ignored at once. 



His words are : — " Mr. Jerrard's paper in this Journal for 

 May 1861 does not disprove a rational communication between 

 the resolvent product and its fifth power. His equation (e\) is 

 not illusory but identical, and we may expunge it and substitute 

 u b for v in the second of the equations marked (e). The result 

 of this substitution, reduced by means of (U), may be made to 

 take the form 



X 5 -f X 4 u + Ag u 2 + • • + X u b = ; 



and the six linear, but not homogeneous, equations 



X 5 =0, X 4 =0,..\ =0 



will determine the six quantities v 5 , v 4 , . . v ." 



Now why is (e^) not illusory but identical ? In the paper of 

 mine here referred to, I arrived at (e ]} U) ; by a slightly different 

 route from that pursued in art. 5. " Combining," 1 said, " the 

 equations (e x ), (U), that is to say, (e^), (U), we find 







the equation (e^) being, as we see/ illusory." Accordingly he 

 finds it necessary for his argument that (e^) should be not illu- 



