OF TRK ROOTS OF SYSTEMS OF EQUATIONS. 
503 
<^10 — ^105 
o 
c,o = (20) h,Q + (10=^) 
c,, = (n) h,,+ {W0i) V>oi> 
Co3 = (^) h,, + (^2) ho„ 
In the expression of c^rj each partition in brackets ( ) has the biweight pq, and 
each partition is attached to a product of quantities such that each factor corre¬ 
sponds to a single bipart in the partition. 
Hence on proceeding to form a relation 
Pi ?% 
^'Ih'h •••=••• + P . . . -h . . . 
Pi P2 ^ ^ 
wherein P represents the complete symmetric function cofactor of ... it is 
clear that P is a linear function of symmetric function products, each of which has a 
specification 
■ • •)> 
and is also a separation of the separable partition 
(ri6-y‘9qy/^ . . .). 
The partitions {piq{' ' ■ • •)’ • •)> separations 
which present themselves in the linear function P are of the same biweiglit. 
When the separations in the function P are all expanded into a sum of monomial 
symmetric functions, each of the latter has the same biweiglit. 
Taking the separable partition r.^s.p , . .) as fixed, a definite number of 
specifications appertain to the separations. Forming then c products in correspond¬ 
ence with each of these specifications, the law of symmetry indicates that the same 
nimber of different monomial symmetric functions will appear in the developments of 
the several linear functions P. Further, the partitions of these monomial symmetric 
functions will be, in some order, identical with the several specifications of the 
separations of the fixed separable partition. 
Assume the specifications to be, in any order 
^1, 02, .. . 0h 
and write the identity 
^2 Pi Pi 
^Vi'U ••• = ••• “h P . -f • . . 
