1909-10.] The Theory of Persy mm etric Determinants. 
413 
Sylvester, J. J. (1851, October). 
[On a remarkable discovery in the theory of canonical forms and of 
hyperdeterminants. Philos. Magazine , ii. pp. 391-410: or Col- 
lected Math. Papers, i. pp. 265-283.] 
The consideration of the problem of the canonisation of the binary 
quintic led Sylvester to the more general problem of determining the p s- 
and q s in 
(Pi x + + (P2 X + WT +1 + + (Pn+i x + 2n+d/) 2n+1 
so as to make this expression identical with 
a 0 £ 2n+1 + (2 n + 1 )a 1 x‘ 2n y 1 4- J(2 n + 1)2 napi^^y 1 + ....+ « 2 n+i2/ 2n+1 • 
This is at once seen to depend on the solution of the peculiar set of 2^ + 2; 
equations 
*1 
+ 
7r 2 
+ . . 
. . + 
TTn+l ~ 
a o \ 
+ 
7T 2^2 
+ . . 
. . + 
^n+lKi+l = 
a l 
+ 
7r 2 X 2 2 
+ . . 
. . + 
^n+lAn+l = 
a 2 
^f +I 
+ 
7T 2 A 2 2 W+1 
+ . . 
. . + 
_ \'ln+\ — 
7r n+l / Si+l — 
a 2n+l 
where the new unknowns ir 1 , 7 r 2 , . . . . , ir n+1 , X 1 , X 2 , . . . . , A n+1 are in- 
troduced merely for shortness’ sake, namely 
7 T r for ip l+1 and A r for q r -r-p r . 
Taking n + 2 consecutive equations beginning with the first, and eliminating 
the 7 r’s, there is obtained 
1 
1 
. . . 1 
a o 
a 2 . . 
• • 'bi+l 
a l 
V 
a 2 2 . . 
' • • a 2 +1 
«2 
A“ +1 
A” +1 . , 
• • • a&; 
a n + 1 
which, if division by the difference-product of the A’s be effected, gives 
a n+l ~ 'S. + ^ 1^2 — • • • • = 6. 
A similar result is evidently reached by taking any n + 2 consecutive 
equations, so that altogether we shall have 
a n + 1 
«n+ 2 
^n+3 
CL>n-x-\ 
a n+ 1 
a n + 2 
n 
+ a n _ x - . 
+ - * 
+ a n+1 — . 
Aj + a. ln _ 1 ^AjAg - . 
0 
0 
0 
l) 
> 
