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DX1(X1a2As...)

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Originally appearing in Volume V01, Page 627 of the 1911 Encyclopedia Britannica.

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DX1(X1a2As...)

DX1(X1a2As...) _ (a2xa...), while D,(ai)sa3•..) =o unless the See also:

PARTITION

partition (81X283...) contains a See also:

PART

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 See also:

law of operation of the operators D upon a monomial symmetric See also:

FUNCTION

function is clear. We have obtained the See also:

EQUIVALENT

equivalent operations 1+µD1+µ2D2 +µ3D3+... =exppol where exp denotes (by the See also:

RULE

rule over exp) that the multiplication of operators is symbolic as in See also:

Taylor's theorem. d'1 denotes, in fact, an operator of See also:

order s, but we may transform the right-See also:

hand See also:

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 See also:

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 See also:

EXPRESS (through the French from the past participle of the Lat. exprimere, to press out, by transference used of representing objects in painting or sculpture, or of thoughts, &c. in words)

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 See also:

POWERS, HIRAM (1805-1873)

powers s1, S2, SS," Further, we can express da in terms of D1, D2, D2, ... by the same law which expresses the See also:

POWER [WILLIAM GRATTAN] TYRONE (1797-1841)

power function s, in terms of the elementary functions al, See also:

A2(z)

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, See also:

F2(X2+Y2)

f2...fm we introduce a new rootµ we See also:

CHANGE (derived through the Fr. from the Late Lat. cambium, cambiare, to barter; the ultimate derivation is probably from the root which appears in the Gr. «a urmu', to bend)

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 See also:

TERM

term covering every See also:

DISTRIBUTION (Lat. distribuere, to deal out)

distribution of the operators of the type presenting itself in the term. See also:

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 See also:

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,See also:

D2D7 (14) (19) (13)

D2D7(213)(214)(15) =A; every other term disappearing by the fundamental See also:

PROPERTY

property of D,.

End of Article: DX1(X1a2As...)

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