932]
ON SYMMETRIC FUNCTIONS AND SEMINVARIANTS.
293
term multiplied by a linear function (with mere numerical factors) of the arbitrary
coefficients: the expression thus obtained must be identically =0; and we thus find
between the arbitrary coefficients a number of linear relations equal to the number
of terms /(ao)6 5 : these relations are independent; for it is only on the supposition
that they are so, that the number of coefficients which remain arbitrary will be
11-7, =4, agreeing with the number of the seminvariants [d 2 ], [c 3 ], [b 2 c 2 ], [6 e ] ;
whereas if the relations were not independent, there would be a larger number of
seminvariants.
But if, instead of the whole set g (ao) b 6 , we consider a set (g oo d 2 ) or say
(ce x c 3 ) and assume for a seminvariant the sum of these terms each multiplied by
an arbitrary coefficient, then operating as before with A we obtain between the
arbitrary coefficients a number of relations equal to that of the terms D (ce oo c 3 ),
and if this be less by unity than the number of the terms of ce oc c 3 , say if we
have (1 — D) (ce x c 3 ) = 1, then there will be a single seminvariant ce x c 3 . We, in fact,
find (1 — D): (g oc d 2 ), (ce x c 3 ), (d 2 x b 2 c 2 ), (c 3 x b 6 ), each = 1, and thus establish the
existence of the foregoing seminvariants g oo d 2 , ce x c 3 , d 2 x b 2 c 2 , c 3 x b 6 . And similarly
if in any case we have (1 — D) (I x F) = 2 or any larger number, then we have 2
or more seminvariants / x F.
43. It will be convenient to write down at once the system of square diagrams
for the several weights 2 to 16; each of these may theoretically be obtained by a
direct process of calculation such as I exhibit for the weight 10, but the labour
would be very great indeed, and I have in fact formed the squares for the weights
11 to 16, not in this manner but by the MacMahon linkage.
w ~ 2 w = 3 tv = 4 w — 5
0 1 1
d
1
e
1
/
1
b 2
b 3
c 2
1
cd
1
b 4 be 2 b 5
w = 6
9
1
ce
1
d 2
1
c 3
1
c 3 b 2 c 2
w — 7
h
1
cf
1
de
1
c 2 d
1
b 3 c 2 V
d 2
b G
bd 2 be 3
