[ 457 ] 



LXII. Note II. on some Objections of Mr. Cayley and Mr. Cockle. 



By G. B. Jerrard*. 



1. [N the October Number of this Magazine Mr. Cayley and 

 -i- Mr. Cockle renew their objections. The former of 

 these mathematicians there says : — " Mr. Jerrard, in his paper, 

 * Note on some Objections of Mr. Cayley and Mr. Cockle/ in 

 the September Number, p. 193, concludes, so far as relates to 

 me, as follows : — e The curious irrelevancy of Mr. Cayley' s objec- 

 tion 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 of u = a, v = a 5 , in virtue of the 

 equation (e^)?' ;; 



The last sentence, I may just observe in passing, was by me 

 put in italics. That the additional emphasis thus given to the 

 point in dispute was not quite needless we shall see presently. 



2. It will be remembered that 



_«=/M/(^)/(' 3 )/(' 4 ), _ 



and that the equation of definition for/(t n ) is 



f(t n ) =x 1 + i n x 2 + t 2n x 3 + t 3 % + t An oc b ; 

 in which x lt % 2 , . . x b denote the five roots of 



x b + A l z 4 + A< 2 x 3 + .. +A 5 =0, 

 and i, P, l 3 , c 4 the imaginary roots of the binomial equation 



p 6 -l=0. 



The function here represented by a is a remarkable one. It 

 possesses, as is well known, the surprising property of having 

 the same number of values as its fifth power. If, however, we 

 resolve a and a 5 into their component parts, 



m, m> /(' 3 )> m, 



[/MP, imT, [/(' 3 )P, LA* 4 )], 



and compare one of these parts, say/(t), in a with the correspond- 

 ing part, [/(*<)] 5 , in a 5 , we shall see that the homogeneity of a 

 and a 5 is not such as to penetrate and pervade these functions. 

 Extrinsically a and a 6 are homogeneous ; but intrinsically — at 

 least in relation to some of their corresponding parts — they are 

 not so. 



As for u, v, (e\), everything relating to them is, in my paper 

 for September, explained thus : — 



i.f " Let us suppose, in conformity with my previous notation 



* Communicated by the Author. 



t i. ii. iii. iv. are substituted in the text for 2, 3, 4, 5 respectively in 

 the original paper, to avoid the clashing of two sets of numerals belonging 

 to the same system. 



Phil. Mag. S. 4. Vol. 24. No. 1G3. Dec. 18G2. 2 H 



