OF THE ROOTS OF SYSTEMS OF EQUATIONS. 
511 
(2) Every partition of the biweight ])€[. 
The right hand side of this relation may be broken up into fragments in each of 
which all the numbers tt^q, 77 ^,. . . . . . are constant. 
In fact we may write 
pi q 
— y 
77io! 77oi! 
Ui7i- • • • 
wherein, following the summation sign Sp, the numbers ttj^q, . . . 77 ^,^,^^ . . . are 
constant, and the operator 
Sp (10'’“ 01'’“ . . . ^ (10^10 +Pio 0l’"oi + Poi _ _;^yiJi5,i + Ppi5i _ J, 
is one of the fragments above mentioned. 
This operation has a biweight 2^<1, and may be defined also in regtird to the partition 
(10’"“0P“ . . . . . .) of the biweight 
51. Write then for brevity and convenience 
Sp (10 “ OP" . . , • •) ^ (icTio + Pio oi^oi + Pol _ _ + Ppiii _ j oI"’“ .. ...) 1 
SO that we may write 
p\ q 
’’’lo- '^oil • • • '^ihh • • • 
(lo’^io ox^oi _ _ 
where the summation is in regard to every partition (10’"“ 01’'“ . . . . . .) of 
the biweight pq. 
52. In general not every partition of the biweight pq will occur in the given 
separable partition, but it is convenient to consider the general result just written 
down as including every such partition. It will be seen later that this result is of 
great importance in the theory. 
I remark that on the left-hand side we have a linear partial differential operation 
whose expression by means of the fundamental symmetric functions and their differen¬ 
tial inverses is w^ell known by what has preceded. Such expression is all that is 
needed so long as we are concerned only with the fundamental forms which, as they 
appear in the expression of a monomial symmetric function of biweight pq, present 
themselves in products which are separations of the symmetric function (10^ 01''). In 
the present broader theory in which the leading idea is the consideration of any 
partition at pleasure of the biweight as the separable partition, we bring into view the 
exhibition of the operation as a linear function of operations, each of which is in 
correspondence wdth a partition of the biweight. 
