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



Again, eliminating in like manner j» 1X1,^92X5, .. .j9„+i\„+i be- 

 tween the 2nd, Srd, ...(» + 2) th equations, we obtain 



fln+2— fln+i2\i + &c . . . 4:^2X1X5 . . .X„+i; 



and proceeding in the same way until we come to the combina- 

 tion of the (n + l)th . . . (2n + l)th equations, and writing 



SX, =*i 



&c. 



2X,.X2 . . . X„+l = *n+I, 



we find 



On^X^a^S^ + an^X'S,^. » » ±ai,8n^\ =0 



«»i+2 — fln+l.«l + «^.*2 • • • +^2'*«+l — ^ 



flfi+S — «n+a«*l4-«n+l.S2 . . . ±«3.«n+l =0 

 &C. &C. 



Hence it is obvious that 



{x-^X^y){x-\-\^) . . . (a:+X«+,.y) 

 is equal to the determinant 



On+i; ««; dn-l', •.. «i 



flfn+2j fl5n-ij fln-2^ ••• ^fg ^* 



• • • • • 



Hence X,, Xg, . . . X„+i are known, and consequently 



are known by the solution of an equation of the (n + l)th degree. 

 Thus suppose the given function to be 



IxAiii ==(Pi^-^9iy)^-^{P^+Qi!/?+(P^ + 93'y^)> 

 we shall have, by an easy inference from what has preceded, 



{Pi^ + QiV) {P^ + 9^) [P^ + W) 

 — a niunerical multiple of the determinant 



* These equations in their simplified form arise from the ordinary result 

 of eUmination in this ease contaming as a factor the product of the dif- 

 ferences of the quantities Ai, As* . i » An+ 1. 



