276 
ON THE THEOREM OF THE FINITE NUMBER 
[761 
There are three squares R 1} viz. these are the squares 
1 
1 
1 
1 
1 
1 
1 
1 
1 
1 
1 
1 
representing the before-mentioned invariants (a — /3) (7 — 8), (a - 7) (¡3 — 8), (a - 8) (¡3 — y) 
respectively: say these are a, b, c, and every other invariant is a rational and 
integral function of these; in fact, the 0-equations give easily 12 = 34, 13 = 24, 14 = 23, 
so that the general form of the invariant is = a 12 b 13 c 14 , where 12, 13, 14 are each 
of them a positive integer number (which may be = 0). Or, what is the same thing, 
the square R e (0=12 + 13 + 14) is a sum 
= 12 . + 13.22/ + 14. R", 
with positive integer coefficients 12, 13, 14, say for shortness it is a sum of squares 
R 1 . And so any like expression with a negative coefficient or coefficients may, for 
shortness, be called a difference of squares R x . 
Observe that, in general, two squares R e , R$ are added together by adding their 
corresponding terms, the result being a square R g+ ^; similarly, if each term of R^ be 
less than or at most equal to the corresponding term of R g , then (but not otherwise) 
the square R$ may be subtracted from R e , giving a square R g -$. 
In the case of the sextic 
(x — ol)(x — /3) (x — y)(x — 8) (x — e)(x — £), 
there are fifteen squares R 1} which may be represented as follows: 
12.34.56 
12.35.46 
12.36.45 
13.24.56 
x 1 
Vi 
z, 
13.25.46 
13.26.45 
14.23.56 
2/2 
^2 
x z 
14.25.36 
14.26.35 
15.23.46 
2/3 
*3 
Xi 
15.24.36 
15.26.34 
16.23.45 
16.24.35 
16.25.34 
2/4 
x 5 
V5 
Z 5 5