ALGEBRAIC Auxiliary Theorem.—The coefficient of in the product
is is where
is a separation of \left( l_1^{\lambda_1} l_2^{\lambda_2} l_3^{\lambda_3} \cdots \right) of specification \left( m_1^{\mu_1} m_2^{\mu_2} m_3^{\mu_3} \cdots \right)
Hence the products s4, s^, s|, s2sj will appear. From the formula (21") ^ 5(2x2) — S(2i)S(i) — -S(2)S(12) + 2S(2)Sfl) S — s — S S 21,1, 2 + s2sI4
3 X
2
and the sum is for all such separations. To establish this observe the result. ixp_^ (srwir^r* Eind remark that pi 3 TTjI^lTTg! (3)jri(21),r2(l3),r3 is a separation of (37ri2,rn,r2+3,r3) of specification (3P). A similar remark may be made in respect of J_y/m —x^2 — y^s ... [J^l Vli’ fj^l ™<i’ and therefore of the product of these expressions. Hence the theorem. Now log (1 + /j.X1 + /*2X2 + y?X3 + ...) = S log (l + /J.a.1x1 + fJ?alx2 + /J?alx3 + ...) a. whence expanding by the exponential and multinomial theorems a comparison of the coefficients of /j.n gives n+v2+r3+...-l (y1+y2 + y3+...-l)! Vivvrs m2(-) y^lvslZ Xn^nf'n,2, xvi+r2+r3+... -1 (l/j + ^ + Tg + ... - 1)! Y k1v v2Y v3
a result easy to verify. Theorem of Reciprocity.—If X^X^... =... + 6^Slhl..)xx^x where 0 is a numerical coefficient, then also X£X£X£... =... + 0(mf ...)<142<3... + ....
jl'-jl'-j*'■■■■ 1 2 3 where since (m^ m^ m^ ...) is the specification of (J 1)-'i(J2)j'2(Js)'73- • • j +... —j1 +y2 +y3 + .... Comparison of the coefficients of iU,i ^ x^x^Xn,,-■ • therefore yields the result , n + ,/2 + >'3+---(>'1-l-^{n 2+^3+... ( )N — >
and another by putting a;1=a:2=a;3=... = l, for then Xm becomes hm, and we have
We have found above that the coefficient of xf^xi^xi2... in the product X^X^... is Mx.^3. ... >
the sum being for all separations of (Z*1/*2^3...) which have the specification (m^m^m^3...). We can multiply out this expression so as to obtain a series of monomials of the form 0(s°ls*2sT3...). It can be shown that the number 9 enumerates distributions of a certain nature defined by the partitions (m^m1^2...), (s^s^2...), (l^l^2...) and it is seen intuitively that the number 9 remains unaltered when the first two of these partitions are interchanged (see Combinatorial Analysis). Hence' the theorem is established. Putting a:1=l and »j=a;3 = a;4=... =0, we find a particular law and, by the auxiliary theorem, any term XJ^X^X^... on the of reciprocity given by Cayley and Betti, right-hand side is'such that the coefficient of x^x^x^... in (1«h)^i(1m2)M2(1’«3)M3_.. =... + d(s°ls*2s*3...) + ..., 3 ^ Y^Y^Y^ is (1si)V1(1s2)<r2(1s3)<r3 .,_... 4.e(m^m^m^...) +... ; mlMo!^!... mx ™2 m3”-
- + ^'(<1<2<3-) + — KSS!‘ •• = -+ 0m^m^m^...) + .... Theorem of Expressibility.—“ If a symmetric function be symbolized by (y.v...) and {f^K3...), (vp>2v3...)... be any partitions of X, p, v,... respectively, the function (pv...) is expressible by means of functions symbolized by separations of (X4X2X3.. .pifj.2/u.3.. .iqi'oVj...). JlW-Js'- — for the expression of Set” in terms of products of symmetric For, writing as before, functions symbolized by separations of (w%»%^3...). X^X^...=SS0(^2X^-O41<!!43>
Let (n)a, (n)x, (»)x denote the sums of the nth powers of
=2?x^x%x$3...,
quantities whose elementary symmetric functions are a1} a», a3, ... ;
xv x2, x3, ; X1} X2, X3, ... respectively : then the result arrived P is a linear function of separations of (Z^1^2^3...) of specification
at above from the logarithmic expansion may be written
(mfmfmf...), and if X^2X J... =SF^^3..., F is a linear
(n)a(n)x=(n)x,
exhibiting (n)x as an invariant of the transformation given by function of separations of (l^l^l^3...) of specification (s^s^s*3...).
1 2 3
the expressions of Xp X2, X3 ... in terms of aq, a;2, a:3, ... ._
The inverse question is the expression of any monomial sym- Suppose the separations of (l^ !^ ^ -..) to involve k different
specifications
and
form
the
k
identities
metric function by means of the power functions (r) = sr. We
have just seen that sr is expressible in terms of symmetric
^ZxZxZ---^s)xhxhxt-- (s=1«2’ •••*>’
function products symbolized by separations of any partition
(r^V-...) of r.
where (m^m^m^3*...) is one of the k specifications.
3
The law of reciprocity shows that
Let this expression be denoted by s^M/Vg ...).
Theorem.—It can be shown that
P(s) ^ZUm^m^m^...),
-1
where
^7rn-l)!(S7r21-l)! - s s s ...7ru! 7r12...! 7t21! tt.^I ... (Ji) (J2) (J3)'
denotes a separation of ■ • ■) an(i the summation is in regard to all such separations. If (Ji), (J2), (Ja) -- he of weights ij, i2, i3,... it is clear that the product • • will appear on the righthand side, and that (i^i^3...) is a specification of a separation of (p^P^Ps3 ■■■'>• 2 Tlx. gr. To express (21 ) in terms of power functions. 2 The separations of (21 ) are (i.) (212), (ii.) (21) (1) ; (iii.) (2) (l22), 2 (iv.) (2) (l) and the corresponding specifications (4), (31), (2 ), (212).
viz. :—a linear function of symmetric functions symbolized by the k specifications ; and that 9st~92 ts. A table may be formed expressing the k expressions PW, P( *,...PW as linear functions of the k expressions (™^1Sm^2’m^3,,")> 5=1, 2, ...k, and the numbers 9st occurring therein 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 ...) whose partition is a specification of a separation of the function symbolized by (l^H^l^3...) is expressible as a linear function of symmetric functions symbolized by separations of (V'^%3--i) an^ a symmetrical table may be thus formed.” It is now to be remarked that the partition ...) can be derived from
