1906-7.] Dr Muir on Axisymmetric Determinants. 141 
considered as a function of the n-\-r letters 
**1 > *®2 > * * * ’ *^ n+r 
divided by the square of the determinant 
^11 ^21 • • • 
dyi 6^22 • • • 
ft lr a 2 r • • • «rr J • 
As regards this we have to remark (1) that again no proof is offered, 
and (2) that the discriminant of U + L 4 x n+1 + . . . + L,.a?, l+r is easily got 
by “ bordering ” the discriminant of U. Taking the case where U is 
ax 2 + by 2 + cz 2 -f dw 2 + 2 fyz + 2 gzn -f 2 hxy + 2pxiv -f 2 qyw + 2 rziu 
with the discriminant 
a h g p 
h b f q 
g f c r 
p q r d , 
and where the linear equations serving to eliminate x , y from U are 
fX Y X + jU-2?/ + poZ + = 0 
VjX + v 2 y + v 3 z + v 4 w = 0 , 
we have, according to Sylvester, the discriminant of the altered U 
equal to 
a h g p ^ v i 
h b f q p 2 v 2 
g f c r v 3 
p q r d [x 4 v 4 
Pi P2 /q /q • 
*1 v 2 V S V 4 * 
l/V 2 | 2 . 
Sylvester (1852, July). 
[A demonstration of the theorem that every homogeneous quadratic 
polynomial is reducible by real orthogonal substitutions to the 
form of a sum of positive and negative squares. Pliilos. Magazine , 
(4) iv. pp. 138-142 : or Collected Math. Papers , i. pp. 378-381.] 
What is really proved here is the important proposition in the theory 
of orthogonants regarding the reality of the roots of Lagrange’s deter- 
minantal equation, or, as it was then called, the equation of the secular 
