OF THE ROOTS OF SYSTEMS OF EQUATIONS. 
523 
In that case and g^^ were shewn to annihilate all functions in which the biparts 
10, 01, respectively, were absent. Hence, expressing all such functions in terms of 
^io> ®oi> • • • have at once a number of symmetric functions of difference of 
the quantities in the identity 
(1 + “h A?/) (1 ~h “h Ay) • • • (1 + ^n'X' 4- A.v) 
= 1 + na^^x -I- na^gj + . . . + 
{n — 'p — cj)\ 
71 . The differential equation 
17io + 9o\ — 0, 
is, as a particular case, satisfied by the solutions of the simultaneous equations 
f/io = 9 q\ — 
In correspondence therewith we have functions composed of differences — «/, 
A — Aj but not of differences — A; ~ A- The functions of differences — «/, 
A — A represented by the infinite series of monomial symmetric functions whose 
partitions contain neither of the biparts 10, 01. 
The generating function for the number of biweight pq is 
_ 1 _ 
(1 - x~) (1 - xy) (1 - yp (1 — a'Q (1 - xhj) (1 - xif) (1 - tf). . . 
The remaining lunctions of differences correspond to those solutions of g^Q 4- g^j = 0 
which are not simultaneous solutions of g^Q = 0 and g^-^ = 0. 
Denoting by N any aggregate of biparts from which both 10 and 01 are excluded, 
we have the system of solutions 
(Id N) - (01 N) 
(Id^N) - (To 01 N) + (dl 3 N), 
(10^ + ? N) - (10^+^-’ 01 N) + . . . + (—)?(lT-'dl?N) 4- . . . , 
for on operating with g-^Q 4" 9 qi — Glio 4" the terms destroy each other in pairs. 
Observe that these solutions are of the same weight but not of the same biweight 
in every term. 
The number of solutions of a given weight is given by the generating function 
X 
(1 - ^) (1 - (1 - . . . (1 oPf + ^ ‘ 
3 X 2 
