OF THE ROOTS OF SYSTEMS OF EQUATIONS. 
515 
Observe that separations of this partition cannot be produced bj any other of the 
partition operators which present themselves on the left hand side of the differential 
equation. Hence if the operand satisfies the differential equation 
9m 
it must also satisfy the differential equation 
9 (To’^io ...) 
59. This important theorem may be enunciated as follows :— 
Theorem. 
“ If a function, expressed in terms of separations of a given monomial symmetric 
function, be annihilated by a biweight operator it must also be annihilated by every 
partition operator of that biweight.” 
As regards the calculation of Tables of Separations of Symmetric Functions, this is 
the cardinal theorem. 
As an example of its application I propose to utilise it for the purpose of exhibiting 
the function (31 01) as a linear function of separations of (21 10 01). The law’ of 
expressibility shows this to be possible, for (21 10) is a partition of the biweight 31. 
Remarking that the separation (21) (10) (01) cannot occur in the expression, since 
it is the only separation which produces the monomial (32) when multiplied out, I put 
(3l ol) = A (21 To) ( 01 ) + B ( 2 loT) (To) + C (Id ol) ( 2 l) + D (dlldoT). 
A monomial symmetric function is caused to vanish by means of the operation of 
the biweight operator if no partition of the biweight is comprised amongst its 
parts. In consequence of this, the only biweight operators which do not cause it to 
vanish are and p^gg. Hence all the partition operators of every other biweight 
operator annihilate the function (31 01). It suffices to employ as annihilators the 
two partition ojjerators p'(oT) and gr(^y 
Hence, retaining only significant terms, 
{0(10) + (Ol) 0(10 01 ) + (^) 0(51 io) + dl) 0(2110 01 )} (3l dl) = 0, 
{9(21) + (10) 0(2llo) + (01) 0(2l w) + (10 01) 0(2l Tool)} (31 01) = 0, 
leading to 
A + C = 0, B + D = 0, 
C + D=0, A + B = 0, 
or 
D=-C=-B = A. 
3 u 2 
