OF THE ROOTS OF STSTEAfS OF EQUATIONS. 
509 
the monomial function at the left of the same line. For example, the terms in the 
second column and third line are separations of (20 01) formed according to the law 
of the function (11 10). Also the terms in the third column and second line are 
separations of (11 10) formed according to the law of the function (20 01). Now 
obseiwe that the law of symmetry establishes that the table enjoys row and column 
symmetry. Hence the assemblage of separations of (20 01) formed according to the 
law of (11 10) is equal to the assemblage of separations of (11 10) formed according to 
the law of (20 01). 
47. Hence in general the theorem :— 
“The assemblage of separations of (rp‘/i • • •)’ foi’nied according to the law of 
{Pi^i' . . .), is equal to the assemblage of separations of (ppj'd' • • •)? formed 
according to the law of • • •)•” 
In the particular case considered the equality is 
- (^ Ol) + (^) (dl) = ~ (TT Id) + (IT) (Id). 
To actually form separations of ('J'lSP’ . . .), according to the law of 
{pi<li^ PP 12 ^' . .), the separations of the former must be written down, and also the 
expression of the latter, by means of fundamental symmetric functions. The separa¬ 
tions are then given the same coefficients as the products of fundamental symmetric 
functions which possess the same specifications. 
§ 8. Third Law of Symmetry. 
48, From the relation 
’’‘1 "^2 Pi P2 
^Pi'll • ' * •••"!“ 
is derived the operator relation 
TTj 7r2 
^ Pl<ll ^ M2 • 
and, thence, by the method already employed, 
(d^/^ Tff . . .)3 = . . . + L . . .)i + . . . 
This law of symmetry is of considerable importance and interest, but I do not stop 
to further discuss it.* 
* Vide ‘American Journal of Mathematics.’ “Third Memoir on a New Theory of Symmetric 
Functions,” now in progress in vols. 11, 12, and succeeding volumes. 
