§25] SEMIN VARIANT LEADERS OF COVARIANTS 43 
for r—dp—w gives O dp ~ w ~ l 5 = 0, etc. Finally, we get 5 = 0, 
contrary to hypothesis. 
Theorem. There exists a covariant K of a binary p-ic 
whose leader is any given seminvariant S of the p-ic. 
The covariant K is in fact given by (3), § 23. By (l), 
for r = co-f-1, 
i20“ +1 5 = 0. 
Hence 0“ +1 5 is a seminvariant of degree d and weight 
w'=w-\- co+1 =pd—w-{-1. 
Then dp — 2w'=—{pd — 2w) — 2 is negative. Hence (4), §23, 
follows from the Lemma. Thus K is annihilated by the 
operator (2), § 23. Next, in 
( a -4h 
the coefficient of x" ~ r y r is 
- l fiO r 5- 7 -^-v,(co-r+l)O î - 1 5^- ! {i20 ï -5-r(co-r+l)0:- 1 5}, 
r\ \r — 1)! r\ 
which is zero by (1). Hence K is covariant with respect to all 
of the transformations T n and T' n of § 19. Now 
T-xTfT-x^V: x=-Y, y=X, 
as shown by eliminating £, ??, £i, tji between 
| x= £ — t], | ¿i, | £i=X—V, 
1 y= v, 1 v = vi + ^i, l Vl= Y. 
Since K is of constant weight, it is covariant with respect to 
every St (§ 16). Hence, by § 19, K is covariant with respect 
to all binary linear transformations. 
26. Number of Linearly Independent Seminvariants. 
Lemma. Given any homogeneous isobaric function S of 
oq, . , , , a v of degree d and weight w, where œ=dp — 2w>0, 
we can find a homogeneous isobaric function Si of degree d and 
weight w-\-1 such that üSi^S.