(lλ
1 lλ
1...) and it is seen intuitively that the number θ remains unaltered when the first two of these partitions are interchanged (see Combinatorial Analysis). Hence the theorem is established.
Putting x1= 1 and x2 = x3 = x4 = ... = 0, we find a particular law of reciprocity given by Cayley and Betti,
(1m1) t(1(1 n1 2) �2 (1.3)�3... = ... +ti (Si 1S2 ?S3 3 ...) -f -..., (PO v1(1s2)a2(1.3)v3... _ ...+o(mi
and another by putting x i = x 2 = x3= ...' =I, for then X. becomes hm, and we have
h,�,,ih,�,,2hm3... _ ... +tir (S? 1 S 2 2 S 3 3 ...) +..., ?1 ?2 ?3 _ � l �2 �3 h S h S2 h 83 ... -. +o (m l m2 m3) +...,
Theorem of Expressibility.—“If a symmetric function be symboilized by (A�v...) and (X1X2X3..�), (�i/-12�3���), (v1v2v3...)... be any partitions of X, respectively, the function is expressible by means of functions symbolized by separation of
X1A2X 3. � � / 1111-2113. � � P1 v2 v3...)”
For, writing as before, Xm 'Xm 2 Xm '= zzo(SQls:2s73...) xi'x12x13..., 1 2 3" 1231 2 3 = EPxi l x A2 x A3, P is a linear function of separations of(/ 1 / 2 A2 / 4 3 3 ...) of specification (m"`1m�2m"`3...), and if X; 1 X 3 2X8 3 ' .. = ?P'xilx12xi 3
P' is a linear function of separations of (li'12 2 13 3 ...) of specification (si 1 s 22 s 33)
Suppose the separations of (11 1 13 2 1 3 3 ...) to involve k different specifications and form the k identities
¿½1s � s Al A 2 A3 .. Xm1sXm2sXm3s... = EP x tl x t2 x t3 ... (S - 1 , 2, ...k), where (m�lsm"`2sm"`38...) is one of the k specifications.
The law of reciprocity shows that p(s) = zti (m 1te2tmtL3t) t=1 st It 2t 3t viz.: a linear function of symmetric functions symbolized by the k specifications; and that () St =ti ts. A table may be formed expressing the k expressions Pa l), P(2),...P(1) as linear functions of the k expressions (m"`'sm�2sm�3s...), s =1, 2, ...k, and the numbers BSc occurring therein is 2s 3s possess row and column symmetry. By solving k linear equations we similarly express the latter functions as linear functions of the former, and this table will also be symmetrical.
Theorem.—The symmetric function (m �8 m' 2s m �3s ...) whose is 2s 3s partition is a specification of a separation of the function symbolized by (li'l2 2 l3 3 ...) is expressible as a linear function of symmetric functions symbolized by separations of (li 1 12 2 13 3 ...) and a symmetrical table may be thus formed." It is now to be remarked that the partition (/,A.1/2)1/42/A38...)can be derived from (m"13m�2sm"`38...) 1 2 3 is 2s 3s by substituting for the numbers mi., m 231 m 331 ... certain partitions of those numbers (vide the definition of the specification of a separation).
Hence the theorem of expressibility enunciated above. A new statement of the law of reciprocity can be arrived at as follows: Since.
P(s) _ /ll8!/12s!/23s!...
t =1 (J1)Jl (J2)?2(J /3... ots(mlllsmtA2sm�3s...), j1 !j2 j3... ls 2s 3s where tist =tit8. Theorem of Symmetry. - If we form the separation function (J2) j1!j2!13!...
appertaining to the function (li'l32l3...), each separation having a specification m" ` ' 8 m �2s m �38 multiply b P (is 2s 3s .��), P Y by ls! /t2s! / 38 !... and take therein the coefficient of the function (mi t tm7t t m 31 t ...), we obtain the same result as if we formed the separation function in regard to the specification (m� It t'tm2 32tm"`l3t...), multiplied by Alt!! /let! /1 3 1!... and took �1a � therein the coefficient of the function (mis m� 2s Es m 3s 3s ...).
Ex.gr., take (li 1 l2 2. ..)=(214); (m ?88m288...) = (321); (m ?i t m2L t...)=(313); we find (21)(12)(1)+(13)(2)(1) =...+13(313)+..., (21) (1)3=...+13(321)+...
The Differential Operators.—Starting with the relation
(1 + a i x) (1 +a 2 x)... (1 +a n x) = 1 +a 1 x+a 2 x 2 +... +a�xn
multiply each side by I +px, thus introducing a new quantity A; we obtain (1 +a1x) (1+a2x)...(1 -Fanx)(1+,ux) = 1+(a1 +1a)x + (a2+1aa1)x2+... so that f (al, a 3, a3,.�.an) =f, a rational integral function of the elementary functions, is converted into f(a1 +12, a2+ p a1,... a n +I la n -i) = f+/ldlf +?`id2f ` `3 d3f+... ?. 1 1 where laan and di denotes, not s successive operations of d1, but the operator of order s obtained by raising d l to the s th power symbolically as in Taylor's theorem in the Differential Calculus.
Write also s l d1= D, so that
f(a i a2+ p al, ...an+Ilan-1) =f +FLDif +F4 2 D2f + t i 3 D 3 f -}-....
The introduction of the quantity p converts the symmetric function 1 2 3 into (XiX2X3+...) -Hu Al (X 2 A 3 .-) +/l02(X1X3.�.) +/103(A1X2.�.) +....
Hence, if f(ai, a 2, ...a n) _ (?i?2%?3���), 1 2 3 +,01(X2A3...) +02(X1X3.�.) +IlA'(XlX2.�.) +... (1 +/-lD1+Fl2D2+�3D3+...) (X i X 2 X 3 ...) � Comparing coefficients of like powers of A we obtain DX1(X1X2X3...) = (X2X3...), while D 8 (X 1 X 3 X 3 ...) =o unless the partition (X3X3X3...) contains a part s. Further, if DA 1 DA 2 denote successive operations of DA 1 and DA2, DX1DA2(x1X2X2...) (%3...), and the operations are evidently commutative.
Also D n D n 2 D;3 (,,{{,,11*1,/,?*2,/,Tr3) = I, and the law of o eration of the p2 X13 ... ['2 3 ... p operators D upon a monomial symmetric function is clear.
We have obtained the equivalent operations
1 +/lDi+ p2 D2+/ 13D 3 - F ... = expμd1
where exp denotes (by the rule over exp) that the multiplication of operators is symbolic as in Taylor's theorem. di 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 as = a08+ aiaas+i+a2aas+2+....
It has been shown (vide " Memoir on Symmetric Functions of the Roots of Systems of Equations," Phil. Trans. 1890, p. 490) that exp(mldl +m2d2+m3d3+...) = exp (Midi +M2d2+M3d3+...), where now the multiplications on the dexter denote successive operations, provided that pp t exp(MiE+M2 2+M3E3+...) +mlH+m2V+m3S3+..., being an undetermined algebraic quantity.
Hence we derive the particular cases 1 1 expel ' =exp(d1 -2d2+5d3 - ...); exp/ld 1 = exp(Ad1p2d2 +/13d3 - ...), and we can express D. in terms of dl, d 2, d 3 ,..., products denoting successive operations, by the same law which expresses the ele mentary function a s in terms of the sums of powers s l, s 2, s3,...
Further, we can express d 8 in terms of Dl, D 2, D3, ... by the same law which expresses the power function s, in terms of the elementary functions a 1, a2, a3,...
Operation of 'D.' a Product of Symmetric Functions. - Suppose f to be a product of symmetric functions f i f 2 ...f m . If in the identity f =f l f 2 ...fm we introduce a new root A we change a 8 into a8+μa8_l, and we obtain
(1 +AD1 2 D2+... +AsDs ...) p Di p2 D2+... -} p3D8 ...) fl X (1 +/lDl+�2D2+...+Asps+...) f2 X.
X (1 +PD1+12D2+...+�8D8+...) fm, and now expanding and equating coefficients of like powers of μ
D 1 f - Z(Difi)f2f3. ..fm , D2f =I(D2f1)f2f3�..fm+2(Difi)(D1f2)f3...fm, D 3 f =F(D3f1)f2f3... f m +Z(D2f1) (Dif2)f3...fm+Z(D3f1) f 2 f 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(1)f.
D1f = D(1)f.+D
D1f = D(1)f.+D