On Quadratic Functions. 415 



and let Y' and Z' respectively become Y" and Z". Eliminate x 

 between Y" = and Z" = 0, and the i-esult is 



AB'-A'B = 0, 

 a simple cubic in y, whence y is known. And x is then given 

 linearly either by Y" = 0, or by Z" = 0, and why /3'= 0, and finally 

 V by /3 = 0. It would not be difiicult to show that when X"=Y" 

 = Z" = 0^ we also (for the same values of the unknowns) have 

 U=V = W = 0. Hence the given system is solved without the 

 occurrence of an equation higher than a cubic. 



Triilitj' College, Cambridge, 

 January 28, 1851. 



LVII. Note on Quadratic Functions and Hyper determinants. 

 By J. J. Sylvestek, M.A., F.R.S. 



To Richard Taylor, Esq. 

 Dear Sik, 



PERMIT me to con-ect an error of transcription in the MS. 

 of my paper " On Linearly Equivalent Quadratic Func- 

 tions " in the last Number of the Magazine. The theorem given 

 at pages 300, 301, and marked (3.), should read as follows : — 



r69m+I bem + 2 • • • bem+s'\_ 

 <.0(l)m+i 0(pm-i-2 • • ■ Oipm + s^ 



{ffll a^...am. ag,^! «0to.^2 • • • ''9m+s <"»+! •««+2 • • • «a+m~» _^ 

 «1 a^. . . Om 0(p^+i ^,pm-{-2 • • ' ^<p,n-'ta ''^"■■'rl «a+2 • • • dn+mJ 

 {Ui a^ . . . Om \^ 

 ilnfl fin+2 • ' • (l„ + ,„J 



I may take this opportunity of mentioning, that by extending 

 to algebraical functions generally a raultiliteral system of umbral 

 iiotation, analogous to the biliteral system explained in the paper 

 above referred to as applicable to quadratic functions, I have suc- 

 ceeded in reducing to a mechanical method of compound per- 

 mutation the process for the discovery of those memorable forms 

 invented by Mr. Cayley, and named by him hypperdeterminants, 

 which have attracted the notice and just admiration of analysts 

 all over Europe, and which will remain a perpetual memorial, as 

 long as the name of algebra survives, of the penetration and 

 sagacity of their author. 



I am, dear Sir, 



Yours very truly, 



Lincoln's-Inn-Fields, J, J, Sylvester. 



April 1851. 



