414 On Three Simultaneous Quadratics of the Fifth Order. 



by bodily suffering. Yet all failed to arrest that brilliant course, 

 which was only to end with his life. Judged by what he has 

 given to the world, he has established a title to the respect and 

 admiration of after times. Still it is to be hoped that that Avorld, 

 which he in life so instructed and adorned, will, now that he is 

 no more, yet have such further advantage from his labours as is 

 to be derived from the publication of those which he may have 

 left recorded. But, even with that advantage, neither posterity 

 nor his contemporaries — except those who had the happiness and 

 the honour of his friendship — can judge of the grandeur and 

 force of the conceptions which teemed in the fertile chambers of 

 his mind, and of the fine qualities which, in the minds of that 

 favoured group, must ever encircle his memory with an enduring 

 halo of affectionate admiration. Considering how imperfectly 

 developed many of his views were, and the little hope that exists 

 of any one fully following out the same career of inquiry, I regard 

 his loss as irreparable. I hope to be pai'doned for this digression, 

 and I now proceed. 



Let U = 0, V = 0, AV=0, be three simultaneous quadi'atics 

 involving the five unknowns v, w, w, y, z. Let \, fx, v, p be four 

 disposable multipliers, and let 



U + XV + yLtW = X, 



U + vV=Y, 



U + pW=Z. 



The expression X is in general of the foi-m 



X = a*^4-/3a7 + 7, 



and /3 is a linear function of v, w, y and z. Let X' be what X 

 becomes when /3 = 0, and let v be eliminated from 7 by means of 

 that linear equation. We may then make 



X' = a«2_^-„y_,.^/y^y_ 



By means of /S'=0 (which is a linear equation in w and z) let 

 w be eliminated from 7', which is a quadratic function of w and 

 s; call the result 7", and make 



X" = «a.'2 + «y + /. 



Now, X" = can be satisfied without determining either x or 

 y. For, we may determine \ and /a so as to make « and a' 

 vanish simultaneously, and we have then only to solve 7"=0, 

 which is, in my nomenclature, a simple quadratic (in z alone). 



By means of /3 = 0, /8' = 0, a;nd 7" = 0, let v, w, and z be re- 

 spectively eliminated from Y and Z. Call the results Y' and Z', 

 then we shall have 



Y' = a'c8 + A.r + B, and Z' = S'.i'2-l-A'.r + B'. 



Determine v so as to render 8 = 0, and p so as to make 8' = 0, 



