284
ALGEBRAIC
FORMS
by substituting for the numbers mv, ... so that we are only concerned with the successive performance of certain partitions of those numbers {vide the definition of the linear operations. For this purpose write = 3«s + ax3aJ+1 + specification of a separation). 2 + ••• • Hence the theorem of expressibility enunciated above. A It has been shown {vide ‘ ‘ Memoir on Symmetric of new statement of the law of reciprocity can be arrived at as the Roots of Systems of Equations,” Phil. Trans. R. Functions S., London, follows :— 1890, p. 490) that Since ea;p(mxdx + m2d2 + mfl3 +...) = eccp(Mxdx + M2d2 + M3d3 +...), tj/.v , . , ^(Ji)h(J yvJ.s) .. PW = Ws! i^ n3s... > - -y , •2 , • , 5 where now the multiplications on the dexter denote successive where is a separation of of specifica- operations, provided that cxp{Mff + M2$2 + M3£3 +...) = ! + mff + mf2 + mf3 + ..., tion ...), placing s under the summation sign to denote the specification involved ; f being an undetermined algebraic quantity. Hence we derive the particular cases MW^)h{jsys... 1 A !*! Me*! Ms* ii-A-isexpd1 = exp{di - L2 + L3 -...); 8 (Ji)A(J2)/2(J3)4... Mis^.Mai^Ma* •), exp[xd1 - ■- exp{ydy - ^M2d2 + ^3^3 -•••), Mie! Med Ms* '•••2 idieiis! • =2^K and we ean express Dsin terms of <fx, d2, d3, .».., products denoting where 0st=6ta. Theorem of Symmetry.—If we form the separation function successive operations, by the same law which expresses the 2(J1)-h(J2WJ„y'3... . . x A A elementary function as in terms of the sums of powers sx, s2, s3, — appertaining to the function Further, we can express ds in terms of Dx, D2,1)3, ... by the same law which expresses the power function ss in terms of the eleeach separation having a specification multiply mentary functions ax, a2, a?t hy Mi*!Me*! Ms*! ... and take therein the coefficient of the function Operation of D, upon a Product of Symmetric Functions.—Supwe obtain the same result as if we formed the pose / to be a product of symmetric functions/x/2.../m. If in the identity /=/x/2- • ffm we introduce a new root y we change at separation function in regard to the specification ...), into as + m®*-x, and we obtain multiplied by Mid Mad Ms;! ... and took therein the coefficient of (1 + /DX + ^22D2+... +/4sDg+...)/ 1 33 the function (m^’m^m * ...). = (1 + mDx + m2D2 + ... + m*H + ...)/ x (1 + /iDx + m D2 + ... + ,usD4s + .. .)/2x 2 4 tl<, t2s A!a:.yr.,take(/V -) = (21 ); (m' mj ...) = (321)
...) x • u = (31s); we find ^ x (1 +//X)x+/a2D2+ ... +^SDS+ ...)fm, 2 3 3 (21)(1 )(1) + (1 )(2)(1) 3 =... +13(31 ) +..., and now expanding and equating coefficients of like powers of y (21)(1) =...+ 13(321) + ..., Dx/=2(Dx/1)/2/j.../m, The Differential Operators.—Starting with the relation D2/= S(D2/x)/2/3. . fm + S(Dx/x)(Dx/2)/3.. ,fm, 1 n (1 + axa:)(l + apc)...{ + anx) = 1 + ape + ape +... + anx I^s/— 2(D3/x)/2/3.. fm + 2(D2/x)(Dx/2)/3. . .fm + 2(D3/x)/2/3.. fm, multiply each side by 1 + yx, thus introducing a new quantityJ M : the summation in a term covering every distribution of the we obtain operators of the type presenting itself in the term. (1 + a1a:)(l + a2a:)...(l + a„a;)(l + px) = + {a^P- /j.)x + {a^ fm1)x‘1 + ... Writing these results so that/(aj, a%, a3,...an)=f, a rational integral function of the Di/= Dq)/, elementary functions, is converted into ^2/= D)(2,/ + D(X2)/, I (3)/+ T>(21)/ + D(x3)/, /K"tMi a2 + M^d••■an + f/Mn-j) —f + ydjf + ^dif+ ^df + ... we may write in general where D.s/=2D(plJ,2p3...)/, j1 v . ai O + <^2+ [-...+ «n-i 5—the summation being for every partition (pxp2p3...) of s, and ~daJ da. oct^ octn 4 4..5.fm. 3 ■ ■ •)/ being = S(Dp1/x)(Dp2/2)(Dp 3/3)/ and d denotes, not s successive operationsth of dx, but the operator D(MiM2M Ex. gr. To operate with D2 upon (213)(21 )(1 ), we have of order s obtained by raising dl to the s power symbolicallyJ as 4 5 3 4 5 in Taylor’s theorem in the Diff. Cal. ■D(2)/= (13)(21 )(1 2 3 )5 + (21 )(1 3 )(1 3 ),4 T>(p)f= (21 )(21 )(1 ) + (21 )(21 )(1 ) + (212)(214)(14), s Write also —d1 - - Ds so that and hence s! <J 2 3 D2/= (214)(15)(13) + (213)(15)(14) + (2134)(2122)(145) + (213)2(14) f{ax +Mj«2 + M i) ...an + m«„-x) =/+ mDx/+ m D2/+ m D3/+ .... + (21 )(21 (1 ). The introduction of the quantity y. converts the symmetric Application to Symmetric Function Multiplication.—An exfunction (XxX2X3...) into ample4 will explain this. Suppose a 3 3 4 we 5 wish to find the coefficient (XiX2X3 +...) + /x i(X2X3 ...) + ...) + yW...) + .... of (52 1 ) in the product (21 )(21 )(1 ). Write Hence, if/(ax,a2, ...an) = {W...), (213)(214)(15) =... + A(524)(l3) +... ; Al A3 (XxX2X3...) + M (X2X3...) + //2(XxX3. ..) + m (XxX2 ...)+... then = (1 + mDx + m2D2 + m3D3 + _ )(XxX2X3!..). D6D4D3(213)(214)(15) = A; Comparing coefficients of like powers of y we obtain every other term disappearing by the fundamental property of Ds. Since Da1(XxX2X3. ..) = (X2X3...), D5(213)(214)(l5) = (13)(14)(14), le I)5 lX ) () 1 eSS the artition we have:— ^“ . pFurther +-if" w A ^ P (W ..) contains a J 3 4 3 D|D®(14)(1 )(1 and^ Da ’ lf •DaiDa2 denote successive operations of Dax 3 )3 = A 4 3 2 D«D? {(13)(1 )(1 )} = A DaxDa2(XxX2X2. ..) = (X3..,), 2 3 )(1 2 )2 2(1 )(1 D D? {5(1 )(1 )(1 ) + 2(143)(12)(1) + 2(143)(13)(1)} 3= A2 2 2 and the operations are evidently commutative. D2D? {12(13 )(1 )(1) + 7(1 )(1)(1) + 2(1 )(1) + 6(1 )(1 )} = A 3 D?12(l) = A, Also D/jJ-DpPpg • • ■ (px • •.) = 1, and the law of operation of the operators p upon a monomial symmetric function is clear where ultimately disappearing terms have been struck out. W e have obtained the equivalent operations Finally A = 6.12 = 72. The operator <7x=aq0ax + ax0a2 + a20a3+... which is satisfied by 1 + mDx + m2IX + m3D3 + ... = expyd1 every symmetric fraction whose partition contains no unit (called where exj> denotes (by the rule over exp) that the multiplication by Cayley non-unitary symmetric functions), is of particular importance in algebraic theories. This arises from the circumof operators is symbolic as in Taylor’s theorem. ds1 denotes, in fact stance that the general operator an operator of order s, but we may transform the right-hand side Xoao0«x + X]a,0a2 + X2a20a3 +...
