140 
Proceedings of the Royal Society of Edinburgh. [Sess. 
by the quantity 
al 2 + bm 2 + cn 2 - '2a mn - '2b' In - 2c Im .” 
Now the said four-line determinant is not resolvable unless A', B', C' be 
changed in sign, and even then the second factor is not as printed, but is 
- ( al 2 + bm 2 + on 2 + 2a mn + 2b' nl + 2c'lm ) . 
The theorem, in fact, may be viewed as giving an expression for the 
product of a ternary quadric by its discriminant; and at a later date 
might have been written 
x y z 
a h g 
= - 
x y j z 
9 
X ; 
h b f 
x A 11 G 
f 
y 
9 f c 
y H B F 
c 
2 
2 G F C 
a h 
h b 
9 / 
No proof is given by Sylvester; but in a footnote we are told that it 
depends on “a theorem given by M. Cauchy, and which is included as 
a particular case in a theorem of my own, relating to compound 
determinants.” What theorem of Cauchy’s is thus referred to it is not 
easy to say. One would think that the most natural proceeding would 
be to show that the coefficients- of x 2 , l y 2 , . . . on the one side are identical 
with those on the other; and, in this case, the names to be mentioned 
would be Lagrange and Jacobi. 
In a postscript Sylvester enunciates a theorem connected with the 
linear transformation of an n-ary quadric; and as this concerns the 
“ determinant ” of the quadric, or what a year later he named the 
“ discriminant,” it necessarily involves a property of axisymmetric 
determinants. His wording (p. 281) is : — “ Let U be a quadratic function 
of any number of letters x 19 x 2 , : . . . , x n , and let any number r of linear 
equations of the general form 
a lr x x + a. lr x 2 + . 
+ a nY x„ § 0 
be instituted between them ; and by means of these equations let U be 
expressed as a function of any n — r of the given letters, say of 
x r+1 , x r+2 , . . . , x n , and let U so expressed be called M. Let 
Ct lr X ^ -p a^X-i • • • H" Q'nr'^n 
be called L,. . Then the determinant of M in respect to the n — r letters 
above given is equal to the determinant of 
U + L i ||| fl + LjjCt‘,,^2 + • • • + \j r X n+r 
