OF THE ROOTS OF SYSTEMS OF EQUATIONS. 
495 
and gpq causes every other single bipart function to vanish; it must, indeed, cause 
any monomial function to vanish which does not comprise one of the partitions of 
the biweight 'pg amongst its biparts. 
21. The relation just obtained yields the equivalence 
Up'i — ( ) 
lA q\ 
i'P + ?-!)' 
and further results of the nature 
9pi'h9ih'h ( ) 
pi + P 2 + qi + q-i 
_ Pi! gdiq! q.J- _g g 
(Pi + — 1) ! (P2 + '73 — 1)! W 
+ 
(Pi + Ps)! ( 7 i + '73)! g 
(Pi + P2 + S'! + 5Z2 — 1 ) • + 
which are of use in connexion with the theory of function with single biparts. 
Since every symmetric function is expressible in terms of the fundamental symmetric 
functions, every operation gpji^'‘ is necessarily expressible as a sum of G products 
and can be performed upon a monomial symmetric function. 
22 . The solutions of the partial difPerential equation 
9p'l — 
are the single bipart forms omitting Sp,^ (Art. 21), while the solution of the partial 
differential equation 
Gpq = 0, 
are those monomial symmetric functions in which the bipart p'y is absent (Art. 13). 
23. The operation is expressible by means of the operations gpp 
Eeversing the formula 
9pi — + «oi 4 ... + 4 . • •, 
'P,l+1 
we obtain 
— 9p? ^109 p + \,2 ^^0i9p,2 + 1 4 • • • 4 ( y ^^hrsf/p^r q + s • 
where as before (Art. 11), 
rYnU Pi P2 
= 2 ..... 
Pl- P2- 
