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ALGEBRAIC FORMS and this, when compared with the assumed relation, establishes monomial 0S, in the development of the assemblage of separations that in any formula connecting the coefficients Cpq and bpq, we are Fgr, is equal to the coefficients of the monomial 0r in the developat liberty to substitute for and for bp(1; we thus ment of P0S. By solving k linear equations we obtain obtain a corresponding operator relation ; further, it is similarly proved that, in any relation connecting the quantities cpq and apq, 0i — Mn Pgj + M12IA + . ■ • + Mi*P ek > we may substitute T>pq for cpq and T)"pq for apq. 02 = M21P $l + MeePft, + • • • + M2*P e*, First Law of Symmetry.—We are now in a position to establish 0* = M*iBe1 + /x*2Pg2-l-... + M**IA, certain laws of symmetry which appertain to the systems of algebraic forms under consideration. The relation (p^)2 = (^5')i and the determinant of this system is also symmetrical since is a necessary consequence of msr = mrs. We can evidently {■pq) shows that any symmetric function in brackets 2 may be Ur8=Psr two symmetrical tables in connexion with every partition expressed as a linear function of products, each of which is a form of a multipartite number. function ( ^ multiplied by a function, and that, moreover, such The last system of equations involves an important theory of expression is symmetrical in regard to the brackets 1 and . expressibility ; for any one of the monomial functions 0 is expressed by a partition which is the specification of some separation Hence we may write of (risi,>1 r<is‘f2- ■ ■ )<l ai (risiPl r2S2P2‘ • •), =... + J(«A a2bf*... . •) and this implies that the parts of the partition of 0 can be + J(«Aai .. )(p]?i7ri Ptff2)! + • • ■ (A) partitioned into^ parts the aggregate of which is identical with Moreover, any product of the coefficients cpq can he expressed as the parts of (j’R'A r2sf-...). Hence the theorem of expressibility. a linear function of terms, each of which contains a monomial ‘‘If the parts of the partition of a monomial function 0 be partifunction of the quantities cq, A a2, /32,... and a product of co- tioned in any manner into parts, wl^h when2 all assembled in a single bracket are represented by AsA rfsf ...), the symmetric efficients bpq, so that we may assume function 0 is expressible as a linear function of assemblages of separations of the function c c ai mi plh • • • + h(«161 ..)6^s1^r^s2- •• + ••• (B) As/1 r2s/2...).” Ptflf1-• • )b%b9rH.(C) We may write the relation {pq={pq){M)i in the form and it will appear that L = M ; for from (B) can be derived the operator relation . :—77 “ 1 ( V 1 ) ! TTj 7r2 ( r 2 -> Tiq ! 7r2! ...As/tW ’l ^ „ Pi P2 D' D' ( + L(aAai«2&2a2-")D" D" Pill Pi h rjq r.^ bllX2... and performing each side of this equation uponai the opposite side of equation (A) we obtain, after cancelling («A aff*), If we express the quantities cpq in terms of the quantities bpq and LD"r1s U"r2S ...(Ahpl^Sp2-A=JD?ri I)7' a i > ft > a2 > > &c-> the symmetric function products which multiply ll 22 Pill Pi'bz or wil 36 a ^PiQifl'ii '' 011 i i ? B them, necessarily separations L=J. 2 an( °f (ihSAlW ---)* i the result of comparing the cofactors of By a similar process, in which equation (C) replaces (B), we find M = J and hence L = M and the equations (B) and (C) indicate l>pqbplfh-• •. on the two sides, will be the expression of (pq) by with L = M a law of symmetry. We may say that in (B) the means of separations of {piq^p^qf^...). Let interchange of the partitions {p^qf1 p^qf2...), {cQbfi off?...) (Ji/W2leaves the number L unchanged. To explain the nature of the theorem that has been established, recourse must be had to the be any separation of a given partition of (pq) ; the comparison notion of a separation of a partition of given specification ; the yields the result definitions of these terms have been given above in respect of a / v27r-l(27T-1)!— single system of quantities and the analogous definitions, in (JrftCftr <-> qTqrr(«)=2(' respect of several systems, will be easily understood without further remarks. Writing the relation (B) in the form the summation, on the right, having reference to every separation TTl TTo c/Pl c/p of the given partition of (pq). This result, when applied to one c PiQiPz,ti -... +i ^Ybr1s1br2-S2 ■ + ... part symmetric functions oin systems of quantities, is the furthest Pl p where P denotes the complete cofactor of b iSb %p.. we may state generalization of Waring’s formula connected with sums of powers that P is a linear function of symmetric function products each of that has yet been made. The inverse formula, for the expression of any monomial function by means of one-part functions, mayJ be which is a separation of written (As/i r2s/2...), (- )Xir~1(Pi(h1TlP2<l‘f'2---) and has a specification 1 A' - 1 ) ! (-7T,-1)!... J k ZS jl j2 ...TTjj !7T12 ! ...7T21 !7T22 ! ... Jl J2”-’ This appears at once by actually forming the product c^e^... where from the separate expressions of cpift, c^,... Let d1,62)...6k he the different specifications which appertain to the separations (J l)j<J 2)ji- ■ • = ilhl'ln11 Pv/h”12-■ ■ )Jl(i32rf217r21A2'?227r22-•■ft--of Tqs/i r2s/2...) and let P0 denote the symmetric function pro- is any separation of (ppl-f1 Ptff2---), the summation is in respect duct, of specification 6, above alluded to. It may be seen that of all such separations, and sj denotes the one part function of when 1^, P02,...P0fc are multiplied out, so as to be exhibited as a linear function of monomial functions, the partitions of the the same weight as the monomial (J). latter are all drawn from the series 0l9 and we may write Second Law of Symmetry.—The operation A 7r2!...wMi IW"’ Bflj = TOl A + TO1202 + ... + mu0k, the multiplication of linear operations being symbolic, and
 * :m

B02 = 21^1 + »i22#2 +... + m^6k, (MAiW2-.--) being a partition of (pq), may he said to have the weight pq ; if the operand be of the same weight the operator Be*=+ w*2^2 +... + mkk0k, is clearly equivalent to the quantities m being numbers. 1 ( d ^i/ d Nn-a he aw s mmetr 7r1!ir2!...UaMl/ damJ - T, i y y, that has been found, may now be stated in the form and thence Vlrs — Wlsr J v ?: i —1 viz., the determinant of the above relations is symmetrical. We 7T|! TTo!.. .n Ptil Pili aPiiiaP2fi2 " Pl may say, in regard to any monomial function (rA n^A...), that Now, from the three identities I., II., and III., we have if 0!, 02,--A be the monomial functions, whose partitions are the specifications of the various separations, the coefficients of the (ma m/2- - )2=...+nphbr^