DX1 (X1a2As...)  _ (a2xa...), while D,(ai)sa3•..) =o unless the partition (81X283...) contains a part s . Further, if DAiDA2 denote successive operations of DA1 and Da2, Da1Da2(X1a2a2...) _ (X3...), and the operations are evidently commutative . Also DPiDmDPB..•(p;1p2ap33...) =1, and the law of operation of the operators D upon a monomial symmetric function is clear . We have obtained the equivalent operations 1+µD1+µ2D2 +µ3D3+ ... =exppol where exp denotes (by the rule over exp) that the multiplication of operators is symbolic as in Taylor's theorem. d'1 denotes, in fact, an operator of order s, but we may transform the right-hand side so that we are only concerned with the successive performance of linear operations . For this purpose write a1 =a°,+a1aa,+1+a2aaa}2+ .... It has been shown (vide " Memoir on Symmetric Functions of the Roots of Systems of Equations," Phil . Trans . 189o, p . 490) that exp(midi+m2d2+m3d3+...) =exp(M1d1+M2d2+Msda+...), where now the multiplications on the dexter denote successive operations, provided that exp(M1f;+M22+M3r,3+...) =1+mlt+mzt;2+mat3+...; being an undetermined algebraic quantity . Hence we derive the particular cases expol = exp(di —zd2+5(13 —...) ; expadl = exp (µdl -2µ2d2+3µ3d2 —...), and we can express D. in terms of d1, d2, d3,..., products denoting successive operations, by the same law which expresses the elementary function a, in terms of the sums of powers s1, S2, SS," Further, we can express da in terms of D1, D2, D2, ... by the same law which expresses the power function s, in terms of the elementary functions al, a2, a3, ... Operation of Da upon a Product of Symmetric Functions.—Suppose f to be a product of symmetric functions flf2.. f .

If in the identity f =f,

f2...fm we introduce a new rootµ we change as into a1+/as—1, and we obtain (1 +µD1+µ2D2+ ... +µ'Da+...)f = (1-}-µD1+112D2+.:.+µ'Da+...)f1 X (1 +µD1+µ2D2+...+µ'Da+...) f2 X . X (1+11D3+112D2+•••+µ'Da+•••)fm, and now expanding and equating coefficients of like powers of µ D1f =~(D1f1)f.fa...f D2f =x(D2f1)f2f3...f +(Dlfl) (D1f2)f3... fm, D3f =E(D3f1)f2f3...fm+ (D3f1)(Dlf2)f3...fm+(D3f1)f2fs...fm, the summation in a term covering every distribution of the operators of the type presenting itself in the term . Writing these results D1f = D(nf, D2f = D(2)f+Du2)f, D3f = D<3)f+D(21)f+Da3)f, et B.,(mµl,mµuemµst, ) 1, z, a, we may write in general De f =ED(plp2p3e.).f, the summation being for every partition (plp2p3...) of s, and D(ptp2p3...)f being =E(Dpl.fl)(Dp2.f2)(DP3f3)f+...fm . Ex. gr . To operate with D2 upon (213)(214)(15), we have D(2)fy' = (13) (Z14) (15) +(213) (14) (15) . Dc12).f = (122) (2P) (15) +(213) (213) (14) +(212) (214) (14), and hence D2f = (214) (15) (13) + (213) (15) (14) +(213) (212) (15) + (213)2 (14) +(214) (212) (14) . Application.to Symmetric Function Multiplication.—An example will explain this . Suppose we wish to find the coefficient of (52413) in the product (213)(214)(15) . Write (213) (214) (15) _ ... +A(524) (13) + ... ; D,D2D7(213)(214)(15) =A; every other term disappearing by the fundamental property of D, .