Mr. Cockle on Equntiuns of the Fifth Degree. 53 



132, 133, and which may be Corrected in a similar manner. 

 For this purpose let 



then it is, in substance, assumed (p. 190) that the equation 



L';+ K+... U' = l'a:^'+V(.v^"- ^^^r) . (5.) 



is only true for a = n, that is to say, for the root x,,. But the 

 two quantities /' and /" may be taken so as to render this rela- 

 tion true, not only for a = 7J, but also for another value of a, 

 which we may select at pleasure from among the suffixes of 

 the roots. Let this other value be h — 1, then (5.) will hold for 

 the root s„^i, and we have 



l„-i=0; 

 so that it becomes necessary in the group («) to write 1„_2 in- 

 stead of l„_i, and consequently to change (a.) into 



[i,.i„_,]^=y;n«-2)=o^ 



[l,.l„_2],^=/2^(«-2) = V . . . . («'.) 



[l,.L-2]^=/^(w-2) = oJ 

 where /"(<^) denotes a homogeneous function of the «th degree 

 and of b undetermined quantities. Hence we have 



?i-2> 3, or ?j>5, {v'O 



whence the least admissible value of ?i is 6, or, in critical cases, 

 7, and the equation of the fifth degree cannot (to say nothing 

 of the additional difficulty occasioned by the occurrence of 

 critical functions) be solved by means of the processes under 

 our consideration, although those processes enable us to effect 

 certain transformations of equations of degrees higher than the 

 fifth (vol. xxviii. p. 395). 



2 Church Yard Court, Temple, 

 November 18, 1847. 



Postscript.— I venture to hope that the time is now ap- 

 proaching when we may expect a discussion of the physico- 

 mathematical problem connected with the Spheroidal Condi- 

 tion of Liquids. I regret that 1 have not had an opportunity 

 of making an attempt, however humble, at its solution; and 

 I would add that the enunciation of it, contained in a letter 

 (signed S .X.) which I addressed to Mr. Taylor from Great 

 Oakley some two years since, and which will be found at 

 pp. 568, 569 of vol.'xxvii. of this work, may be advantageously 

 modified. ---J. C. 



