Mr. J. Cockle on a Theory of Transcendental Roots. 145 



♦ . . . This, which may be called ' the law of equal transference/ 

 is first to be recognized ; and it is found to prevail in all media, 

 whether regular or peculiar/' 



It thus appears that Mr. Ponton regards "the law of equal 

 transference " and u the semel-bis-ter law 9> as laws of nature ; 

 whereas I am unable to view them as anything else than the 

 immediate results of two arbitrary assumptions which it was 

 necessary for Mr. Ponton to make before he could determine 

 nine unknown quantities by seven equations. 

 I am, Gentlemen, 



Yours sincerely, 

 KfcW Observatory, BALFOUR STEWART. 



June 1860. 



XVIT. Sketch of a Theory of Transcendental Roots. 

 By James Cockle, M.A., F.R.A.S., F.C.P.S. %c.* 



IN these and otherf pages I have, I believe, proved the hope- 

 lessness of solving a quintic by means of an Abelian sextic ; 

 at all events I have given formulae which afford a ready test of 

 any supposed solution. Solubility by radicals is, indeed, a cha- 

 racteristic of equations of the lower degrees, and of certain forms 

 of those of the higher ; but the conclusions of Abel and Sir Wil- 

 liam Rowan Hamilton indicate that it is not a property of equa- 

 tions in general. And if we would study the general problem 

 of solution, we must introduce functions which are not in general 

 algebraic, though in special cases they may be susceptible of 

 algebraical expression. The introduction of such functions is 

 facilitated by results with which Cardan, Tschirnhausen, and 

 Mr. Jerrard have enriched science, and which manifest the pos- 

 sibility of reducing to the form 



x n — nz+(n— 1)# = 



any general equation of a degree lower than the sixth. The 

 possibility of this transformation entails, save in the case of 

 quadratics, that of the transformation 



w n — nacc + a n =0 



within the same limits. The non-symmetric and the symmetric 

 forms alike disclose that the roots are functions of a single para- 

 meter ; and, inasmuch as these forms are substantially as general 



* Communicated by the Author. 



t See my "Notes on the Higher Algebra" in the current (June 1860) 

 Number of the ' Quarterly Journal of Pure and Applied Mathematics/ In 

 a paper " On the Resolution of Quintics," which appears in the same 

 Number, I have given some additional development to formulae discussed 

 in this Journal. 



