§36] 
COVARIANTS IN TERMS OF THE ROOTS 
57 
we see that k equals a polynomial P(ai) whose coefficients are 
rational integral functions of the differences of x/y, ou, . . . , a p 
in pairs. Since 
K{Aq, . . . , A p ; %,7])=K{a 0 ,. . .,a p ;x,y), Ao = ao, rj=y, 
= x(oci, . . . , a p , y ). 
The left member equals P{a!i) since 
P(ol\ —n)— P(ai) = 0 
for every n. Hence a\ does not occur in P{cn), and k is a 
polynomial in the differences of x/y, a\, . . . , a v . 
Let W be the weight of K and hence of the coefficient of 
y w . Then k is of total degree W in the a’s and of degree 
in x/y. Thus 
• X 
K = 'Ec i \product of w differences like —a T 
[ y J 
• {product of W — co differences like a r —a s {. 
Hence 
K = ao d I,Ci{product of w differences like x—a. r y] 
• {product of W — co differences like a r —aff. 
Next, for x=—ti, y = ¿, / becomes ^ = ^0^+ • • • with 
a root — 1 /oc T corresponding to each root a T of /. The function 
K for F is 
i . i 
Ao d ZcA product of co differences like —rj = — 
• | product of W — co differences like 
Using the value of A 0 in § 35, we see that the factor 
(-l) pd ai d ... a/ 
must be cancelled by the — a T and the a T a s in the denominators.