[ 395 ] 



LXV. On the Finite Solution of Equations. 



By James Cockle, M.A., Cantab. : Special Pleader*. 



[The subject concluded from p. 191.] 



15. 1" ET A', a", . , ju, be n unequal integers, then it might 



be shown f that x 9 * equals 



Po +Vx' x>/ + Px" x> ~'+ &c.; .... (ae.) 



and, hence, that J A'" x x> " + A iv x x " may be reduced to the 

 same form (ae.). Consequently its second and third terms 

 will amalgamate, respectively, with the first and second terms 

 of the right-hand side of (a.) (thus becoming unavailable), and 

 its only effective part is 



Po + Px v »** + &c (a£) 



If, therefore, the number of terms in (af.) be < 2, we shall 

 have (sup. p. 132), 



u" = a" 2 p k „, b' = a\ 1 ^ + a l "p^. . . . (ag.) 



16. In general, then, the transformation (b.) of that page 

 can be effected for equations of the fourth degree without 

 the necessity of fulfilling (ag.) ; but in critical^ cases we are 

 limited to the fifth and higher degrees, since p disappears. 

 On this account biquadratics cannot be reduced to a binomial 

 form, as we might otherwise have inferred ||, for in such case 

 we have, ultimately, to satisfy two homogeneous equations 

 between two quantities of the form A + p x . 



17. So, beyond all doubt, the transformation (o.) of p. 190 

 can, in general, be effected for equations of the sixth degree, 

 without satisfying (ag.) by means of one cubic, two quadratics, 

 and five base equations. But in critical cases we meet with 

 the same obstacle as that mentioned in the last paragraph, and 

 are limited to the seventh and higher degrees ; so that the 

 solutions of equations of the fifth and sixth degrees present 

 distinct difficulties T[. If they are absolutely insoluble, may 

 we not hope, from a consideration of the modes in which they 

 evade different proposed methods of solution, to arrive at a 

 more elementary demonstration of the fact than has yet ap- 

 peared ? 



On the Reduction of certain Functions. 



In those cases, in which the length of the calculations is 



* Communicated by T. S. Davies, Esq., F.R.S. and F.S.A. 

 t Sup. p. 191, Note *. + Sup. p. 132. § Sup. p. 191, par. 12. 



|| Sup. p. 133, par. 5. 



IF See Sir W. R. Hamilton's "Inquiry" (cited sup. p. 191. Note *), p. 

 298, line 25, and p. 317 [9.]. 



2E2 



