761] 
OF THE CO VARIANTS OF A BINARY QUANTIC. 
277 
viz. 12.34.56 here represents the square R lf for which the terms 12, 34, 56 (and 
of course the symmetrical terms 21, 43, 65) are each =1, the other terms all vanishing; 
or, what is the same thing, it represents the invariant (a — /3) 12 (y — 8) 34 (e — £) 56 . But 
it is not true that every square R g is a sum of squares R 1 ; this is not the case, 
for the square U 2 , 
= 12.13.23.45.46.56, 
representing the invariant 
(a - /3) 12 (a - 7 ) 13 (¡3 - 7 ) 23 (8 - e) 4S (8 - £) 46 (e - £) 56 > 
is not a sum of squares Rj. 
But the square last referred to is a difference of squares Rp it is in fact 
= 12.36.45 + 13.25.46 + 14.23.56 - 14.25 .36, 
or, what is the same thing, the corresponding invariant is the product of the 
invariants 12.36.45, 13.25.46, 14.23.56, divided by the invariant 14.25.36; viz. 
it is a rational function of invariants R x . 
It is required to show, first, that every square R e is a difference of squares R l ; 
and thence, secondly, that it is a sum of a finite number of squares R* (being, in 
fact, squares R 1 and R 2 ). 
For the first theorem we equate the general expression of R g with the assumed 
value 
x x . 12.34.56 + y x . 12.35.46 + z x . 12.36.45 + ... + z*. 16.25.34. 
We thus obtain 
fifteen equations 
satisfied by 
12 = y 1 + X x + 2-1 
x 1 = 34 — 26 
+ r + s — t, 
13 = x 2 + y a + 
x 2 = 13 — 25 
+ p — r 
+1, 
14 = x 3 + y 3 + z 3 
x 3 = 14 
-p 
-S 
15 = Xi + 3/4 + £4 
x 4 = 15 — 26 — 
36 +p + q + r ■+ s , 
16 = x 5 + 3/5 + ¿5 
x 5 = 45 
— q — r 
> 
23 = x 3 + x 4 + x 5 
3/4 = 12-34 + 26 — q — r 
— s + t, 
24 = x 2 + 3/4 + 3/5 
3/2=2 5 
~P 
p 
25 = y 2 + y 3 + z 5 
3/o = 
P 
) 
26 = z 2 + z 3 + Z\ 
3/4 = 36 
-p-q 
p 
34 = x x + z± + z 5 
3/5=16- 45 
+ q + r 
-t, 
35 = y 1 + 3/5 + z 3 
= 
2 
p 
36 = 3/3 + 3/4 + z i 
^2 = 
r 
p 
45 = x 5 + z x + 
2- 3 = 
s , 
46 = + y x + 3/2 
2^ = 26 
— r 
-8 , 
56 = Xi + x 2 + x s 
Z h = 
t,