396 Mr. J. J. Sylvester on a remarkable Discovery in the 



a b c 

 i.e. i^-j/(ae— 4Z>(/4-3c*)+ bed =0, 



c d e 

 in which equation it will not fail to be noticed that the coeffi- 

 ^cientofv^is zero, and the remaining coefficients are the two 

 well-known hyperdetenninants, or, as I propose henceforth to 

 call them^ the two Invariants of the form 



aa?*H- 4i^y + 6ca!^y^-\-4dan^ + ^ ; 

 be it also farther remarked that 



'=8(59 -2^» 7^^ 



in which equation the coefficient of Sfi is the Determinant or 

 Invariant of 



x^ + Siscy + S2.y^ 

 y being thus found, s^y s^ and fi being given by the equations in 

 terms of v are known, and by the solution of a quadratic X^, \ 

 become known in terms of s^, s^ and/, h in terms of Xp \^ 

 fjL, and the problem is completely determined. The most symme- 

 trical mode of stating this method of solution is to suppose the 

 given function thrown under the form 



4- 6e(/a? -f ^^y) (/^^ +5'V)- 

 Then writing 



—V, the quantity to be found by the solution of the cubic last 

 given becomes 



8€ 



(^»-f) 



I shall now proceed to apply the same method to the reduction 

 of the function 



a^a^ + SdoF . y + 2Sa^y^ + 56agX^y^ + 70a^a^t/* 4- ^6a^a^y^ 



-\-2Sae . xY + Sayxy'^ -\-as.f, 



tinder the form of 



-f 70€(piX + q^yY{p^ + q9y)\Ps^ + q^?{p^x -f q^)\ 

 It will be convenient to begin, as in the last case, by taking 



S'i=i'i^i q%—P'P^ %=P'?^ 94=Pi\ 

 €PiPiPaP4=^> 



