290 Mr. A. Cayley on Equations of the Fifth Order. 



Mr. Jerrard's writings ; but with reference to arts. 7 and 8 of 

 his last " Note," &c, I must say that the restrictions which he 

 seeks to impose on Lagrange's theory of similar functions seem 

 to me unwarrantable. Who would think of superadding a second 

 rational relation, as suggested in art. 8 ? If the value of v given 

 in that article coincides with the value given by (e^), then (e^ 

 may be expunged as superfluous ; if the values do not coincide, 

 then one of the repugnant equations has been improperly intro- 

 duced. In employing Lagrange's theory we may confine our 

 attention to (E x ), and need not concern ourselves about (e^ in 

 any way. 



Felixstowe, by Ipswich, Suffolk, 

 September 2, 1862. 



XXXVIII. Final Remarks on Mr. Jerrard's Theory of Equations 

 of the Fifth Order. By A. Cayley, Esq.* 



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: — "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 dis- 

 cussed, which is this : — Is Lagrange's theory inapplicable to the 

 case of u = u } v = u 5 } in virtue of the equation (e\)?" Rut if 

 my objection be (curiously or otherwise) irrelevant, then the pro- 

 position I contend for might be admitted without prejudice to 

 Mr. Jerrard's results : this proposition is, that Lagrange's theory 

 is applicable to the case of w = a, v = a 5 , which is a case not ex- 

 cluded by the only exception (the case of equal roots) to the 

 general theory, and therefore, notwithstanding the equation (e' x ) 

 or anything else whatever, coming within the general theory. 

 Mr. Jerrard, in his reply, contends for the contradictory propo- 

 sition that, in virtue of the equation (e'j), Lagrange's theory is 

 not applicable to the case u = a, v = a 5 J and he thus in effect 

 treats the objection as a relevant 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 the issue 

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

 it is not my intention to do so. 



2 Stone Buildings, W.C., 

 September 13, 1862. 



* Communicated by the Author. 



