460 Mr, G-. B. Jcrrard on some Objections 



contends for the contradictory proposition that, in virtue of the 

 equation (e\), Lagrange's theory is not applicable to the case 

 u = x, v = a 5 , and he thus in effect treats the objection as a rele- 

 vant one. It appears to me that the objection is not only a 

 relevant one, but that the proposition therein contended for is 

 completely proved ; at any rate " — observe this — u the issue is 

 so narrow a one that it seems useless to argue it further, and it 

 is not my intention to do so." 



By the word objection I have been accustomed to mean, u that 

 which is presented in opposition ; adverse reason or argument." 

 To me therefore it seems as clear as any axiom of Euclid that 

 Mr. Cayley's objection, in which (to use the very words quoted 

 in art. 1) he merely occupies himself in proving what I had taken 

 for granted, while he leaves untouched the main question to be dis- 

 cussed, cannot rightly be classed with relevant objections. No 

 matter what the subject in dispute is, an objection characterized 

 as above is, I hold, essentially irrelevant. By mentally supply- 

 ing the characteristic of Mr. Cayley's objection in every place, 

 the incongruity of his remarks will at once appear. 



Just compare my paper in the Philosophical Magazine for 

 May 1861 with Mr. Cayley's for March 1862. 



In the first of these two papers it will be seen that, after stating 

 generally the results at which Lagrange had arrived with respect 

 to homogeneous functions, and pointing out (e'j) as charac- 

 teristic of the case of u-fifizfizfi**, v^fif^ft^fc 4 ) 5 , I 

 thus conclude : — " We are therefore not permitted to assume 

 that the resolvent product can in general — that is, when U has no 

 equal roots — be expressed rationally in terms of its fifth power. 



"Again, it is generally possible to establish a rational com- 

 munication between that fifth power and the function W t, as is 

 evidenced in this latter case from the non-existence of any illusory 

 equation corresponding to (e\). 



11 We are thus furnished, as will be seen, with a new con- 

 firmation of the validity of my method of solving equations of 

 the fifth degree." 



Here, then, the controversy ought to have ended. In Mr. 

 Cayley's attempt to fasten a contradiction on my method, La- 

 grange's theory had given way. That theory cannot bind (e\) . 

 No stronger confirmation of the validity of the method in 

 question can easily be imagined than this. 



Let us now turn to Mr. Cayley's paper for March. From the 



* The function afterwards denoted by a, called also the resolvent 

 product. 



t The function W depends on the final equation of the sixth degree, 

 discussed in my ( Essay on the Resolution of Equations.' Published by 

 Taylor and Francis, Red Lion Court, Fleet Street, London. 



