454 



Prof. Sylvester on the Theorems 



theorem of Jacobi, given at the end of his well known memoir 

 " De Eliminatione Variabilis e duabus sequationibus," Crelle, 

 vol. xv. p. 101, a very slight inspection of which at once leads to 

 the further and interesting inference that the resultant of the 

 canonizant of an odd-degreed function of x and unity, and of the 

 canonizant of the second differential coefficient of that function in 

 respect to x, is an exact power of the catalecticant of the first 

 differential coefficient of x in respect to the same. This is the 

 essence of the matter communicated by Mr. Cayley ; but subse- 

 quent successive generalizations of the theorem have led me on, 

 step by step, to the discovery of a vast general theory of double 

 determinants, i. e. resultants of bipartite lineo-linear equations, 

 constituting, T venture to predict, the dawn of a new epoch in the 

 history of modern algebra and the science of pure tactic. 



I will begin this note upon a note, by reproducing in brief the 

 first of my two demonstrations of the simple theorem in question*. 

 Let us write 



77 



A Q -1, A,- a h , A 2 _ a b C , A 3 - bc d e 



oca j r 



c a e j 



and so on. And again, let 



A j — a, A 2 — 



A 3 — 



and so on. The theorem in effect to be proved is simply this, 

 that the resultant of X* and X*_i is an exact power of \ it which 

 (as will at once be seen) is the coefficient of x { in X t . In what 

 follows, I shall use R(P, Q) or R(Q, P) to denote indifferently 

 the positive or negative resultant of any two functions P and Q, 

 ignoring for greater simplicity all considerations as to the proper 

 algebraical sign to be affixed to a resultant of two functions taken 

 in an assigned order. 



Jacobi's theorem above referred to, stated so far as necessary 

 for the purpose in hand, is as follows : 



X„=(A*+B)X„_ 1 -^-X„_ 2 . 



n— 1 



Hence, by virtue of a general theorem of elimination f, 



R(X W , X 



n-\)— \n-\^ \ ~~ ^2 X«-2> X n _! ) j 



^w-1 



* The second has been communicated by Mr. Cayley in the March 

 Number of this Magazine. 



t This theorem is best seen by dealing in the first instance with U, V, 

 any two homogeneous functions of x, y of degrees n, n—\ respectively 



