278 Proceedings of the Royal Society of Edinburgh. [Sess. 
will make 
vl + — yl + . . . 4- —y]n — + - 7- £2 + • . • + -^-P n . 
a 2 a n b 1 b 2 b n 
The result is the theorem that — If ike substitution 
} k=n 
7c^l 
changes 
hyl + %2 + • • • +b n yl into a x x\ + a 2 x\ 4- . . . +a n xl 
the conjugate substitution will change 
—yl + —yl + • • • +— yl into . . . + \-x 2 n . 
cii ci n bi b 2 b n 
The rest of the paper is occupied with theorems which hold only in the 
case of four variables. 
Sylvester, J. J. (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. Philos. Magazine, 
(4) iv. pp. 138-142 : or Collected Math. Papers, i. pp. 378-381.] 
The terms “ orthogonal transformation ” and “ orthogonal substitution ” 
date from the year 1852, the former appearing in a paper of Sylvester’s 
published in the February part of the Cambridge and Bub. Math. Journ. 
(see vol. vii. p. 57), and the latter in the title of the paper now reached. 
In the former paper, too, the word “ unimodular,” as applied to a trans- 
formation, is first used (see p. 52), the meaning being that the modulus — that 
is to say, the determinant of the coefficients of transformation — is then unity. 
As has been already noted * when dealing with axisymmetric determi- 
nants, this opens with the proposition that when a rs = a sr , 
a n + x 
« 12 . 
. . CL ln 
a n — x 
«12 • ■ 
■ • a in 
a n 
a 22 + x . 
• • <^2n 
a 21 
C&22 • < 
• • <^2 n 
a nl 
«n2 • 
• • <Lm 4" 
CL n i 
• ■ 
■ • ® nn 
<7u - x 2 q 12 ... q ln 
<721 <Z 22 • • • <h n 
q n i q n 2 • • . qnn-P 
* On verifying this, see also the account of the related paper published in the Nouv. 
Annales de Math, for November 1852. 
