ALGEBRAIC
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and the corresponding operator relation 1 »r2 Pi Pa -7—j—d' +PD" D" 1- 7r2---- Pj-?! P292 r2s2 where P consists entirely of symmetric functions of the quantities appertaining to the first identity. Assuming now a relation
(i.) the expression of dpq in terms of the operations
(ii.) the expression of {pq) by means of elementary functions. It will be seen that the same law obtains. Ex.gr. <fu=:cf(10 01)-<f(ll) <f21=cZ(T02 01) - cf(20 01) - rf(n 10) + rf(2l), are examples of the expressions of weight linear operators in + (W1 ^P2-)2= - + -> terms of partition linear operators. The latter have develoned and employing upon its left-hand side the right-hand side of the expressions such as operator relation and vice versd, we obtain the result P = Q, showing that, in the relation d(10) =-!^ + (10)-£^ + (01)-^J-=-+ (11)+ ..., tf(10) V(102) a(10 01) ' a(ll 10)+ we are at liberty to interchange the partitions (m"1 MJ"1-• •) and (rjs/i Third Law of Symmetry.—From the relation + "mfPA •• + • mic: to which corresponds TTi 7T2 D' D' ... = ... + LD"r D"2 . Pill P2I2 v ^2 we obtain by the usual process (rjs/i m"*-•) + •••; involving an interesting law of reciprocity which has been discussed in the Memoirs referred to. Linear Separation Operations.—A monomial function is expressible as a linear function of separations of any one of the partitions obtained by partitionment of its parts (see ante). In particular the partition (!(/ Ol") can be thus obtained, from any monomial of weight pq, and thence we find that the latter is expressible in terms of the elementary functions, since every product of these, of weight pq, is in fact a separation of (10p01?). The linear operations dpq, as above defined, are suited to operate upon such expressions, but are not at present adapted to operands associated with a separable partition other than (10p 01?). Let the separable partition be c
^(20 ol) = —+ (10)-_^ +.... c?(20 01) d{20 10 01) The alternant (“Combination,” “Zusammensetzung”), of any two partition linear operators vanishes, and, as a consequence, if d{A), d(B) be any two operators and <f> a solution of cf(A) = 0, d(B)<£ will also be a solution of the same equation. It has also been shown {loc. cit.) that if a function, expressed in terms of separations of a given monomial symmetric function, be caused to vanish by any weight operator, every partition operator of the same weight will also cause it to vanish. This is a cardinal theorem appertaining to the expression of any function by means of separations of a given partition. Ex. gr. Suppose it requisite to express the function (31 01) by means of separations of the function (2l 10 Ol); the “law of expressibility ” {ante) shows this to be possible, because (21 01) is a partition of (31). Assume (31 01) = A(2l l0)(01) + B(21 0T)(i0) + C(10 0l)(2T) + D(21 10 01), since the tenns(21)(10)(01) clearly cannot present itself. <f31, dm, dS2 do not make (31 01) vanish, because (31 01) involves partitions of 31, 01, and 32 ; but every other weight operator makes it vanish. Selecting dw and d21 we may make further selection of the partition operators c?(10) and <2(21). Operating then with
+(oi) _ + (21) _ +(21 oi) — —, <2(10) <2(10 01) <2(2110) <2(2110 01) and with -L--Kl0)-^-^.- + (6l)-^ _ +(l0 0l) _—— , and of this let any separation be <2(21) <2(21 10) <2(2101) <2(21 10 01) (Ji)Ws..., we find A + C = B-|-D = C + D = A + B = 0, and thence (Jj), (J2),...being the distinct factors that may occur. Then (31 01) = A{(21 10)(01) - (21 01)(10) - (10 01)(21) + (21 10 01)}. <f„ = Sc?rs(J)-^-r, the summation being for all the factors (J) that a[p) There are many ways of showing that A = - ^ ; perhaps the may present themselves. most instructive is to make use of the relation <231 =... - D31 + ... Let the general expression of (J) be =... + <2(21 10) + ...; and, performing - D31 and <2(2110) on opposite (10Pio+o-io01Poi+o-ox _< ) . sides, then, since -(0i) = A{(0i) + (0i)} or A=-i. , /+s-l(r + s-l)!_, V / r!. s!. It has been seen that a generalization has been made from a 2p-l (Zp-1)! - Pio Poi... rrs/i..., weight operator d.pq to a partition operator. The same thing D D D ii obtains with respect to the obliterating operator T)pq. In the Pio'Poi'---/VlSl! . 10 01 case of a single system of quantities it was established that Dp is we obtain performed upon a product of monomial functions through its r+*-l(r + s_l)! various partitions ; we have, in fact, to pick out all partitions of (-) r-^-dr r s! p in all possible ways from the given product, taking one part (So -1)! _ — — d only from each factor of the product; the component operation — (io<rio01<7°i...r1s/r1s1...)^, ■Pnsy associated with the partition oip we may appropriately the summation being in regard to every distinct factor (J) that denote by may present itself in a separation of the given separable partition, D^rK2--)and to every partition (10Plo01Po1...) of the weight rs of the given So similarly, in respect of several systems, we have the weight linear operator drt. operator and the partition operator Regarding p10, p01, ...as constant write d ..), and we have the equivalence 2 (lO^io Ol^oi.. .riS/riSi..— = d(10Pio oiPoi.. (T Vpq = '2T>{p1qfip2qfri...) and the operator relation becomes the summation being for all partitions of the weight {loc. cit. § 10). r+s-l(r + s_ 1)1 (-) , , dr, Ex. gr. D10 = D(10) ' ' r!s! Du=D(T0 0l) + D(n). 2p 2((2p-l ! _pl _Po: r s P v The connexion between the weight operators dpq, DPJ has been Pio!pn01!...p t /r s !... < c?(lU ° 01 • i i Vi'")’ established ; the corresponding relations between the partition are important. The analogy between quantity and the summation now being in regard to every partition (10pi° OIP01...) operators must be kept in view ; with this object denote by of the weight rs. Observe that dn is a linear weight operator ; 3operation ? ,ri 3 ,r2 expression of Sp, by means of separations of that <f(10Pio01poi...) is a linear partition operator; and that we (i i9'i 1 l 25'2 ”-) the have expressed the weight operator as a linear function of the (PiPS P^-), and by Sp^ the sum of all the monomial sympartition operators of the same weight. Compare metric functions of weight pq. Sp? differs from hpq as will be seen S. I. — 37
