19G Mr. J. Cocldc on the Remarks of Mr. Jerrard. 



equations of the sixth order respectively, a 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 coefficients of 

 each equation are rational functions of the coefficients 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 a can be so expressed seems 

 so clear, that I must apologize for giving a formal proof of it. 

 In fact, if a, f3, 7, 8, e, f are the six values of a, then, denoting 

 by 2 ;i the sum of the nth powers of these quantities, we have 



« + ■ +. 7 + '« ;+ 6 + f=S, 



« 5 . « + /3\ /5 + 7 5 . 7 + S 5 . S+A-e+t* ?=5fl, 



a 25 , a + yS 25 . f3 4- 7 25 . 7 + S 25 . S + e 25 . e + ? 25 . ?= 2 26 ; 

 and thence 



(a 5 -/3 5 )( a 5 -7 5 )(a 5 -S 5 )(a 5 -e 5 )(« 5 ~r°)« = S 26 -S/3 5 .2 21 



+ 2/3y . 2 16 -2/3yS 5 .2 n + 2/3 5 7 5 oV. 2 6 —/8 5 7 5 S 5 e 5 , 



where X/3 5 denotes the sum /3 5 + 7 5 + S 5 + e 5 -f-f 5 ; and in like 

 manner 2/3 5 7 5 , 2/3 5 7 5 S 5 , S/3 5 7 5 5 5 e 5 denote the sum of the pro- 

 ducts of the quantities /3 5 , 7 s , S 5 , e 5 , f 5 , taken two and two, 

 three and three, four and four together. 



But 2 P S 6 , ... S 26 are symmetrical functions of a, /3, 7, S, e, f; 

 that is, they are rational functions of the coefficients of the equa- 

 tion for «, or, what is the same thing, of the coefficients of the 

 equation of the fifth order; and the product (a 5 — /3 5 )(a 5 — 7 5 ) 

 (a 5 — S 5 )(a 5 — e 5 )(a 5 — ty), and the coefficients 2/3 5 , &c. g-wa sym- 

 metrical functions of /3 5 , <y 5 , 8 5 , e 5 , £5, are rational functions of 

 a 5 and of the coefficients of the equation for a 5 ; that is, they 

 are rational functions of a and of the coefficients of the equation 

 of the fifth order. The only case of failure would be if two or 

 more of the quantities a, /5, 7, $, e, J were equal ; but this is not 

 the case, since we are only concerned with the general equation 

 of the fifth order. Hence by the last equation, a. is given as a 

 rational function of a 5 and of the coefficients of the equation of 

 the fifth order. 



2 Stone Buildings, W.C., 

 February 4, 1862. 



XXIX. Note on the Remarks of Mr. Jerrard. 

 By James Cockle, M.A. $c* 

 1. TUT HERMITE'S results are reconciled with the possibi- 

 JLtX • lity of solving binomial equations of the fifth degree 

 * Communicated by the Author. 



