224 Concluding Remarks on a recent Mathematical Controversy, 



issue the solution of the sextic is manifest from the circumstance 

 that the expression 



e\ei+eidl+eiei 



is a rational function of one root only of the quiutic, and conse- 

 quently that the root of the quintic will contain no quintic surds. 

 That the expression in question is such a rational function may 

 be inferred from the foot-note to art. 88 of my paper of March 

 1860, from the third division of my " Notes on the Higher 

 Algebra " in the ' Quarterly Journal of Mathematics/ June 1860, 

 or more directly from my " Supplementary Researches in the 

 Higher Algebra }) in the first volume of the third series of the 

 ' Manchester Memoirs/ Mr. Jerrard has not avoided a diffi- 

 culty which, as he has pointed out in his paper of January 1846, 

 must arise ; nor has he given any definite enunciation of the 

 theorem to which, at least according to his paper of February 

 1846, that of Cauchy must yield its place. On the cover of this 

 Journal for November 1860 Mr. Jerrard announced that he was 

 preparing for the press an Appendix (to his f Essay ') relating to 

 the mode in which cubic radicals enter into the expressions for 

 the roots of equations of the fifth degree. Until that Appendix 

 shall have appeared, I retire from a controversy the continuance 

 of which seems only to elicit from Mr. Jerrard a repetition of 

 refuted arguments. However much I may differ from Mr. Jer- 

 rard on certain points, I repeat my acknowledgements of the 

 instruction and advantage which 1 have derived from the study 

 of his writings. 



Brisbane, Queensland, Australia, 

 June 16, 1863. 



P.S. — The writer desires to add that he hopes shortly to forward a 

 paper extending the correlations, between algebra and the differential 

 calculus, implied in the term differential critical function, to other functions 

 for which he will suggest the name "differential covariants." He also 

 adds that, to the best of his recollection, the first step in the theory of six- 

 valued functions of six letters, commented on in Prof. Sylvester's paper of 

 May 1861, was made, in the Paris Memoir es for 17/1* by Vandermonde, 

 who also, in the paper just alluded to, gave a cyclical process, connected 

 with that the working powers of which have been so admirably developed 

 by Mr. Harley.-— Brisbane, June 18, 1863. 



