44 
ALGEBRAIC INVARIANTS 
In (2), § 24, replace S by IT 1 S, whose degree is d and 
weight is w—r+1, so that its co is co+2r —2. We get 
QO r n r ~ 1 S—O r il r S=r{u+r-l)O r ~ 1 Q r - 1 S. 
Multiply this by 
(-1V- 1 , 
r! co (co —1~ l) . . . (o)+r—1) 
The new right member cancels the second term of the new 
left member after r is replaced by r — 1 in the latter. Hence 
if we sum from r = 1 to r = w+l, the terms not cancelling are 
those from the first terms of the left members, that from the 
right member for r= 1, and that from the second term on 
the left for r = w-\-1. But the last is zero, since ti w+1 S=0, 
STS being of weight zero and hence a power of a 0 - Hence 
we get tiSi=S, where 
w + X 
Sx= S 
(-D 
T — 1 
= ir!w(w + l) 
(w+f — 1) 
O r ü r ~ l S. 
Theorem.* The number of linearly independent seminvariants 
of degree d and weight w of the binary p-ic is zero if pd—2w< 0, 
but is 
{w; d, p)-(w-1; d, p), 
if pd~2wf_ 0, where (w; d, p) denotes the number of partitions 
of w into d integers chosen from 0, 1, . . . , p, with repetitions 
allowed. 
If p=4, (4; 2, /0 = 3, since 4+0, 3 + 1, 2+2 are the partitions of 4 into 
2 integers. Also, (3; 2, /0 = 2, corresponding to 3+0, 2 + 1. Hence the 
theorem states that every seminvariant of degree 2 and weight 4 of the 
binary p-ic, p^A, is a numerical multiple of one such (see the Example 
in § 20). 
The literal part of any term of a seminvariant S specified 
in the theorem is a product of d factors chosen from ao, ax, 
. . . , a v , with repetitions allowed, such that the sum of the 
subscripts of the d factors is w. Hence there are (w; d, p) 
possible terms. Giving them arbitrary coefficients and oper 
ating on the sum of the resulting terms with i2, we obtain 
a linear combination S' of the (w — 1; d, p) possible products 
* Stated by Cayley; proved much later by Sylvester.