OF THE ROOTS OF SYSTEMS OF EQUATIONS. 
517 
Supposing cipq changed into q + vcip, we have from previous work 
(1 + p.Gio + ^^01 +.+ +•••)/ 
= (1 + /rG^o + ^^01 +.+ + . . .)/i 
X (1 + p-Gio + z^Gqi + . . . + fxPv'JGpq + . . .)J.2 
X . . . i 
X (1 + P'G^o + vGqi -f ■ . • • + l^^’l''^Gpry + . . •)fm^ 
Expanding the right hand sid-e and equating coefficients of like products of powers 
fjL and V, we get 
G,of= t (Gio/i) (Gio/a) + t (G20/1) AA . . . f>n, 
Gn/= (Gio/i) {GqJ^A ■••/.. + S {G^Ji)Afs . . . /;«, 
G3o/= 2 (G,o/i) {G,M (G.o/s) /..../« + 2 (G,o/i) (G,o/.) /s ••./. + 2 (G 30 / 1 )/. • •. A, 
G.2J= 2 (Gio/i) (Gio/o) (Go^/s)/^ ...A + t (Gii/i) {G^oA)A • • • 
= 2 (G2o/’i) (GQiy’3)y3... yii + 2 {G^ifi) aA • • • 
and so forth, where the summations are, in regard to the different terms, obtained by 
permutation of the m suffixes of the functions A^ A^ • • • A»- 
In general in the expression for Gpqf there will occur a summation corresponding to 
each partition of the hi weight pq. If a partition be {piqi p^fh • • • summation is 
2 (G„j./i) (G„/,). . . 
Gl. Thus, when performed upon a product of functions, the operator G^,^ breaks up 
into as many distinct operations as the biweight pq possesses partitions. It is 
convenient to denote the operation indicated by the summation 
■S+ 1 
A 
P_ 
(ri2i ■ 
. . Ihq,)) 
and to speak of it as a partition operator. 
62 . We may now write down an equivalence 
Gpq XG(^ ^ . _ . ■^), 
where the summation is in regard to every partition of the biweight pq. This is, in 
fact, a theorem for operating with Gpq upon a product of symmetric functions, and it 
is consistent with the more sumple law previously established. 
