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XVII. Notes on the Resolution of Equations of the Fifth Degree. 

 By G. B. Jerrard, Esq.* 



1. TT is clear that an expression for a root of the general equa- 

 J- tion of the fifth degree must involve radicals characterized 

 by each of the symbols %/ ', V and V. If, however, we examine 

 all the solutions which have hitherto been discovered of parti- 

 cular equations of that degree, wc shall find that into none of 

 them do cubic radicals enter. A great if not an impassable barrier 

 seems at first view to oppose their introduction. For how can 

 cubic radicals arise unless there be a cubic equation ? And how 

 can there be a cubic equation, unless, in opposition to the well- 

 known theorem of M. Cauchy, the number of different values of 

 a non-symmetric function of five quantities can be depressed to 

 three ? I propose now to inquire whether the method which I 

 have given in my " Reflections on the Resolution of Algebraic 

 Equations of the Fifth Degree/' will enable us to solve these 

 questions. 



2. Turning to No. 44, which contains the first application of 

 that method, we find (see this Journal for June 1845, vol. xxvi. 

 p. 572)- 



" The equation of which 



W/'+ 1 R(W/') 



is a root will evidently be of the third degree. For omitting 

 the parentheses connected with 1 R I we sec that 



the exponent, as is usual, indicating a repetition of an operation ; 

 and that consequently the root in question will not be affected 

 by writing f\fie} instead of/\ 

 " We must also have 



(W/ + iRW/)(ab)(cd) . . = (V F )(ab)(cd) . . , 



when (ab)(cd) . . takes the form (ab)(ab) ; but not for all values 

 of a, b, c, d, . . : since the method of continuous substitutions 

 will not generally be applicable to processes based upon the 

 theorem (v, w.), which is, as we must remember, hypothetical in 

 itself. 



" Hence I conclude that there will be an equation of the third 

 degree with given coefficients simultaneous with the equation 

 V 15 + C X Y H + . . =0, which cannot be depressed [by any further 

 equalization of its roots] below the 15th degree without inducing 

 certain relations among A v A a , . . A 5 ." 



I proceed to verify a result apparently so inexplicable. 



* Communicated by the Author. 



