OF THE ROOTS OF SYSTEMS OF EQUATIONS. 
499 
The left hand side of this equation is 
log (1 + ^^10 + + • • • + + • • •) {vide Art. 16), 
while if the operators g" be replaced by their expressions in terms of the operators 
G", it is easily seen that the right side is 
log (14" + AGtoi d“ • • • "h pq +•••)■ 
29. Hence the operator relation 
1 + + • • • + ^pq^^'ff^ + • • • 
= Ids (1 + “h ^s^oi '’?+•••+ "h • • >)• 
This result must be compared with the relation (Art..24) 
1 + ^10^ + CqiTJ + • • • + c,pq^^'ff + • • • 
= ns(l + 4* • • • + ^d‘^sd>pq^^'ef +•••)• 
30. I say that such a comparison yields the following theorem :— 
“ In any relation connecting the quantities Cp^ with the quantities hpr^, we are at 
liberty to substitute 
Gy;^ foi Vyp(p and ^^pq Goi i^pq > 
and we in this manner obtain a relation between operators in correspondence.” 
To explain this further, observe that f, rj being undetermined cpiantities in the 
assumed relation which connects the quautities of the three identities L, II., III., we 
are able to express any product whatever of the coefficients c^q, Cq^, . . . Cp,^, ... in 
terms of products of coefficients 6jo, ^oi> • • • ^pq^ ■ • • of symmetrical functions of 
the quantities a^, /3^, a^, /3.,, . . . The substitution in question can he made in any 
equation thus formed. 
31. With regard to the relation of Art. 24, viz.: — 
1 + + Cfji’? 4“ • • • + Vpij^Pyf^ + . . . 
= dij(1 4" + ^d\nV 4" • • • 4* l^sd^pq^^’g'’ + •■•), 
two important facts have been established— 
(i.) That the relation is unaltered when the quantities occurring in the first 
identity 
1 -}- ci^(pc fi- ciQ^y + CAp,fcPy'‘ + . . . = (1 + age + ^gj) (1 + age -j- ^gy) . . . 
3 s 2 
