OF THE ROOTS OF SYSTEMS OF EQUATIONS. 
507 
Similarly, the operation 
f > 77 ^ ! 
will be regarded as being of weight (biweight) where (. • •) is a partition 
of the biweight ; and if, as a particular case, the operand be of the same biweight 
fq, it will be equivalent to the operation 
< 7 r, ’ TT^ I 
and will be of degree equal to 
+ 7^3 + • • • — 
-TT. 
Since therefore 
we have the result 
assuming then a result 
dff da 
“Pl?l 
TTi! TTg! 
= 1 , 
9p.q:^9pa: 
TTp' TTg! 
— -- • • • — 1 ; 
Pi p^ 
• • •)2 = • • • + '^Ks. K, 
derived from the three initial identities of Art. 24 and the relation assumed to exist 
between the quantities involved, we are at once led to the operator relation 
fi ^2 Pi P2 
••• =.■. + PG'v,.,G",... 
Tr;^ ! TTg ' 112 2 
where P consists entirely of symmetric functions of cpantities which occur in the first 
identity. Further suppose a second result 
. . .)3 == • • • + K 
P2% 
4- 
Hence, operating on the left and right of this result wnth right and left sides of the 
foregoing operator relation, we obtain 
y Pi^h y p-2q 2 • ■ ■ / 
ttP TTa' . . . 
“b Qdyi,r/, 
+ •••)’ 
or from theorems established above (Arts. 13, 44) 
P = Q, 
no other terms surviving the operation, 
45. Hence a theorem of symmetry :— 
3 T 2 
