415 
1909-10.] The Theory of Persy mmetric Determinants. 
As a matter of fact (p 1 cc + g 1 2/)(j9 2 cc + ^) .... {p n+1 x + q n+ 1 y) 
= 2>iP 2 . • -Pn+i(x + \y)(x + X 2 y) .... (x + \ n+1 y), 
= P1P2 • • • Pn+l(x n+1 + 2 A 1 - xn y + 2 A 1 A 2 • xn ~v + ••••)» 
= P 1 P 2 • • • Pn+1 (A 0 x? l+1 + A^y + A'F n ~ 1 y 2 + . . . ), 
^0 
PlP 2 ' • • Pn +1 
A 
A 0 
5 
P1P2 . . . p n+1 
ctn+iV + a n x a n y +a n _ x x .... 
A» 
a n +vy +a n+ i x a n +iV + a n x .... 
a‘ln+lV + a 2 n X CLlnU + <hn-& • • • • 
from which we see (1) the point which Sylvester wished to make, namely, 
that ppc + qpj , p 2 x + q 2 y , ... . being viewed as the original unknowns, it 
is important to know that their values are multiples of the linear factors of 
A, and (2) that A = A 0 (as + X 1 y)(a3+X 2 y) 
Of course the conclusion drawn is that the transformation of a binary 
(2^+l)-ic into -the sum of n -\- 1 powers depends on the solution of a 
determinantal equation of the (n-\- l) th degree. As examples, the quintic 
and septimic are taken, the latter mainly for the purpose of drawing 
attention to the fact that the conditions of “ catalecticism,” that is, of 
{a , b , . . . , h } x , y) 7 being expressible in the form of the sum of three 
seventh powers — instead of four, as the general rule provides — require that 
the cofactors of the elements of the first row of the determinant 
y 4 - y 3 x y 2 x 2 — yx 2, x 4 
a b c d e 
bed e f 
c d e f g 
d e f g h 
must all vanish, or, what by the homaloidal law is the same thing, that 
two of them vanish. 
The analogous problem for even-degreed functions is next taken up, 
a beginning being made with the transformation of the quartic 
{a , b , . . . , e jf x , 2 /) 4 into the form 
(Pi x + W)* + ( P2 X + Wf + 6e ( Pi x + 9i Vf{ P2 X + Wf • 
On putting 
2l = jP]Al > P2 ~ #2^2 > *P\P2 = P > + ^2 = S 1 ’ ^1^-2 = S 2 
there is obtained by equatement of like powers of x and y 
