280 Proceedings of the Royal Society of Edinburgh. [Sess. 
and of course 
Pi 
q 3 
Q 2 
a 
f 
e 
Q s 
p 2 
Qi 
= 
f 
b 
d 
Q 2 
Qi 
*8 
e 
d 
c 
No determinant notation, however, is used, nor are determinants spoken of. 
Hermite, Ch. (1853, May). 
[Sur la theorie des formes quadratiques ternaires indefinies. Crelles 
Journ., xlvii. pp. 307-312 : or (Euvres, i. pp. 193-199.] 
[Remarques sur un memoire de M. Cayley relatif aux determinants 
gauches. Cambridge and Bub. Math. Journ., ix. pp. 63-67 : or 
(Euvres, i. pp. 290-295.] 
In his paper of 1846 Cayley, as we have seen, gave a general solution of 
the problem of the transformation of xf + xf + . . . into + £ 2 2 + . . . by 
means of a linear substitution. Hermite now faces a more general problem, 
namely, “ la transformation en elle-meme d’une forme quadratique quel- 
conque,” a problem which in itself is rather outside our subject, but which, 
by reason of the important modification made in the initial step of the 
solution, deserves attention. 
The quadric being /(aq, x 2 , . . . ), the problem is to find the most general 
linear substitution which will transform 
f( x i 5 X '2 y • • • ) into 5 £2 ’ • • • ) ’ 
and Hermite having before him Cayley’s expressions, in the simpler case, 
for the cc’s and £’s in terms of an intermediary set of variables, and observ- 
ing that any member of the intermediary set is the arithmetic mean of the 
corresponding members of the two given sets, begins by imagining merely 
“ que les quantites x et £ soient exprimes par des indeterminees auxiliaires 
0, de sorte qu’on ait en general 
x r + $ r = 2(9,.” 
There is thus obtained 
/(aq , x. 2 , . . .) = /( 20! - $ 1 , 26, - g 2 , . . .), 
ii/ft,.,, ...) - <<$+«$+...) 
+ A( i>h> • • • )> 
so that in order to have /(aq, x 2 , . . . g 2 , . . . ) it is seen to be neces- 
sary that 
- 2 M> ■ ■ •)• 
