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XX. Remarks on M. llermite's Argument relating to the Alge- 

 braical Resolution of Equations of the Fifth Degree. By G. B. 

 Jerrard*. 



1. "WAS not aware of what M. Hermite had written with 

 JL respect to the impossibility of effecting generally the 

 algebraical resolution of equations of the fifth degree, until I 

 saw an abstract of his argument in a paper by Mr. Cockle which 

 appeared in the last Number of the ' Quarterly Journal of Pure 

 and Applied Mathematics/ The abstract to which I refer is 

 this : — 



"M. Hermite's argument may help to settle a still vexed 

 question. It is as follows : 



" Let us assume that between the roots of the sextic reduite of 

 the general quintic there exist relations which render that sextic 

 an Abelian. These relations would, in effect, lead to the con- 

 clusion that the reduite is resolvible algebraically by quadratic 

 and cubic radicals; and without having recourse to the demon- 

 stration of Abel, we may at once convince ourselves that it would 

 follow that the equation of the fifth degree is resolvible by 

 radicals of the same knd. Let us call x Q} os l} x 2 , x 3 , x 4 the roots 

 of this equation, and put 



U — Xr\X-i ~\~ XyXn ~J~ XqXv ~J~ XoXa ~J~ X aXqj 



V ^^ XqXq "t* * 2 4 ' *4*1 • * V"3 * ™2> Q) 



the quantities u + v and uv will be, the one rational, and the 

 other a root of an equation of the sixth degree resolvible alge- 

 braically by hypothesis. Then u and v and their various values 

 will be expressed by means of quadratic and cubic radicals. The 

 same conclusion will hold with respect to the more general 

 functions 



U a = OVi)" + ( Vs)" + (<%)" + fe* 4 ) a + ( Vo)°, 



v = (x x 2 ) a + {x 2 x 4 ) a + (a? 4 ff ,)« + (a? 1 a ?8 )« + {x 3 x ) a , 

 whatever be the integral exponent a. It follows that 



c^q^j, AyAcy ^2<*3) t* 3 c* ^y ^4^ J 



for example, will satisfy an equation of the fifth degree, the co- 

 efficients of which will only involve radicals of the kind in ques- 

 tion. But with two values of u a and v a it will be possible to form 

 two equations of the fifth degree having a common rooty for 

 example x x v the others being different. Hence we may 

 deduce x x l} and consequently the similar function x -j-x l} in 

 terms of cubic and quadratic radicals ; consequently also x and 

 a?„ so that the equation of the fifth degree would be resolvible 

 without quintic radicals. 



* Communicated by the Author. 



