§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.