~28~-

satisfy the equations ZInyix = 3, *°°*, INxdx = q, and where
=T, $ Mgt =1,

As with Theorem V, we have two corcllaries,

CoroLuarY VIII.1. The regpresentation of Sjp...q in terms of
the ¢'s is independent of the weights,

Corovvary VIII, 2, Let j', k', **° , q' be a persutation of
Js ky **°, @+ The extressions for Sjree.q and for $i1kteueg! -have
the same coefficients, and the indices of one expression may by

obtained from the other by permutation,

Froducts and quotients of infinite products.

We shall now consider the expansion of the product

M= T[1+v'] I[1+v"], which is the product of two infinite

products of the same kind, where v;' and v;" are not derivatives
but are two unrelated v's, Let s' and s" denote the corresponding
s's, and «' and «" the correspcnding u's. Frow (7),
log M = 1ogTI[1+vy'] + leg I[1 4 vg")
A)'(’l). m xh('l)" !

—m—-—s'“.ur + %’E—-Efi—-—l"un.r ’

=Euh' +3 wmh = %Z
where the inper summation is taken for the values of the indices
satisfying the equation ja + B+ ***+rA=h, and where = denotes
jtk+ ***+r~1, as previously defined.

Since the series are power series and therefore absolutely
convergent, coefficients of the same power of x may be combined. Now
we will let y, = v'h +4", ; hence w = <" 'Ep("—ll)’l:-:—'h'(s',k..., H8 dhvea?)
the summation being for indices such that ja + 8 + ***+vA=h . This
is foroula (7) with each s replaced by s' +s", so that throughout
the rest of the work of deriving the formula for Cy the same change
is to be made., Therefore (11) gives C, for M by replacing each s by
s +s" .

It is obvious that (11) gives C, for the infinite product