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XXVI. On a new Class of Theorems in elimination between 

 Quadratic Functions, By J. J. Sylvester, M.A. 9 FM.S.* 



IN a forthcoming memoir on determinants and quadratic 

 functions, I have demonstrated the following remarkable 

 theorem as a particular case of one much more general, also 

 there given and demonstrated. 



Let U and V be respectively quadratic functions of the 

 same 2n letters, and let it be supposed possible to institute n 

 such linear equations between these letters as shall make U 

 and V both simultaneously become identically zero f. Then 

 the determinant of aU + jw/V, which is of course a function of 

 A and (x of the 2nth degree, will become the square of a func- 

 tion of X and ju, of the ?ith degree ; and conversely, if this de- 

 terminant be a perfect square, U and V may be made to va- 

 nish simultaneously by the institution of n linear equations 

 between the c 2n letters J. 



Let now P and Q be respectively quadratic functions of 

 three letters only, say x, y, z ; and let 



U = P+ [lx + my + nz)t 

 V = Q + K(/o? + my + nz) t. 

 The determinant of AU + p/V in respect to x 9 y 9 z 9 1 is easily 

 seen to be (A + ^) 2 x the determinant of 



AP + ju,Q + {lx -\-my-\- nz)t 

 in respect to a?, y 9 z 9 1. Hence if we call 



AP + ^Q + {lx + my + nz)t = W, 

 and make ] | .W a squared function of A, ju, or which is the 



xyzt 



same thing, if 



{W}=o, 



Xf* xyzt 



U and V will vanish simultaneously when two linear relations 

 are instituted between the quantities (all or some of them) 

 x, y 9 z 9 t. 



In order that this may be the case, it will be seen to be 

 sufficient that 



P = Q = (to + my + nz) = 



* Communicated by the Author. 



f In the more general theorem above alluded to, the number of letters 

 is any number m, the number of linear equations being any number not 



exceeding — . 



X When nssl, we obtain a theorem of elimination between two qua- 

 dratics, which has been already given by Professor Boole. 



