OF THE ROOTS OP SYSTEMS OF EQUATIONS. 
497 
and generally in the expression of Cp.^ every symmetric function of biweight pq of the 
quantities in the first identity occurs, each attached to the corresponding product of 
coefficients from the second identity. 
25. Represent the symmetric functions of the quantities occurring in the second 
and third identities by partitions in brackets ( )i, ( )o, respectively. 
Now 
(1 + + ^s^oiV + • • • + + • • •) 
is from the identity II. equal to 
ns[(l + (1 + (•••).••]> 
which is 
n,n,(l + . 
26. Hence the assumed relation becomes on taking logarithms 
ts log (1 + + ^Arj) = tXt log (1 + ’ 
and expanding and equating coefficients of 
(Mh = (pA (pAi ; 
an important relation which shows that the assumed relation is unaltered when the 
set of quantities a is interchanged with the set in such wise that and are 
transposed. It is indeed of fundamental importance, and will be brought prominently 
forward in the sequel. Its consideration must be postponed until a further step has 
been taken in the theory of the operators. 
27. Let the operators 
qpiji 
Ppq ) Gt/)-? 5 
ffpq ? , 
refer to identities I., IL, III., respectively. 
Writing the relation 
1 + ^10^ “k <^uiV + •••■!" + • • • 
= 11^(1 + + AAoiV + • • • + ^JAs^AAP'^ +•••)> 
in the abbreviated form 
U — • • • ) 
3 S 
MDCCCXC.—A. 
