Mr. J. Cockle on Transcendental and Algebraic Solution. 139 

 and, o) denoting an unreal cube root of unity, we may write 



Again, determining E by the condition that the whole series 

 must change sign when a changes sign, and that at the last of 

 the set of real values we have 



x = — 2, or 1, or 1 



in a succession corresponding to that given above, we find 



Consequently the relation 



x m = (<o m - co 2m ) . \f~-L . 2« 2r « 2r + (co m + G> 2 ™)2/3 2r+1 a 2r+1 



will, wheo m is replaced by 0, 1, and 2 successively, give the 

 three values of x. 



The foregoing is a complete illustration of the process in its 

 application to cubics. Quadratics lead to a linear differential 

 equation. In the case of the higher equations, series may be 

 obtained corresponding to that above given, even though radicals 

 corresponding to R t and R 2 have no existence. The increasing 

 complexity of the process as we pass the fifth degree may per- 

 haps be met by the following modification of it. Let there be 

 given 



x n + lx n ~ l + mx n ~ 2 + . . +r=0. 

 Change this equation into 



x n + l\(a)x n - 1 + m[jb{a)x n - 2 + . . + rp(a) = 0, 

 where X, //,, . . p are functional symbols which, a being replaced 

 by unity, or by c, satisfy the respective sets of conditions 



\(1) = 1, Ml) = l,..p(l) = l, 



or 



\(c) = l, /x(c) = l, ..p(c) =1, 

 but which are in other respects arbitrary. Then if, treating a as 

 the independent variable, and /, m, . . r as constants, and apply- 

 ing the foregoing process, we can, by means of the arbitrary con- 

 stitution of \(a), p,{a) } . . p(a), obtain the n— 1 particular inte- 

 grals of the differential resolvent, the n values of x must be 

 sought by writing 1, or c, in place of a in those integrals, introdu- 

 cing arbitrary constants, and pursuing a path already traced in 

 the case of cubics. This modification of a process which I glanced 

 at in the August (1860) Number, would render it unnecessary to 

 deal with more than one parameter. 



4 Pump Court, Temple, London, 

 November 5, 1861. 



