ALGEBRAIC equation are symbolic and unsymbolic respectively, provided that mpq, are quantities which satisfy the relation exp (Mio£ + M0117 + ... + W-pq^Tp + ...) = +mv^ + mo'r)+ ...+mpq^1‘rii+... ; where {, 17 are undetermined algebraic quantities. In the present particular case putting m10 = M, '//i01 = v arid 'Mpq — 0 otherwise M10I + Moit? + ... + + ... = log (1 + MS + "’?) or and the result is thus
exp(fj.dw + vd0l)
= exp {/«fio + ^01 - ^(^20 + +1/2^02) + • • •} = 1 + /uD^q + i'Doi + • • • + Ppv,*&pq + ... ; and thence + vdm - ^(^20 + + ... p q = log (1 +mD10 + »'Do1+ ...+ P v T)pq + ...). From these formulae we derive two important relations, viz. : _ g+g-i(p+ ?-!)! 127r-I(S7r-l)!rtu1 T)7t2 P ! ^..1- ! M2 ^ ^ pq 2(' »+g-l r^i+gi-Diryc^+ga-Dir2 (-) D2 * S-A P^.q^. / pj-qj- -1 TT ,77ri .;7r^ ... TTt ! 7T9. “‘PiQiPM the last written relation having, in regard to each term on the right-hand side, to do with Stt successive linear operations. Recalling the formulae above which connect sprj and a2„p we see that dp,, and Dpq are in corelation with these quantities respectively, and may be said jto be operations which correspond to the partitions (^), (10^ 01?) respectively. We might conjecture from this observation that every partition is m correspondence with some operation ; this is found to be the case, and it has been shown (loc. cit. p. 493) that the operation (multiplication symbolic) % 7T, ! 7To Pi'll PA corresponds to the partition (Piq™1 PM™2-")- The partitions being taken as denoting symmetric functions we have complete correspondence between the algebras of quantity and operation, and from any algebraic formula we can at once write down an operation formula. This fact is of extreme imj stance in the theory of algebraic forms, and is easily representable whatever be the number of the systems of quantities. We may remark the particular result
FORMS
287
This assumes that the coefficients Cpq of III. are certain functions of the quantities and coefficients of I. and II. Denoting symmetric functions of the quantities in I., II., and III. by partitions in brackets ( ), ( Jx, ( )2 respectively, we find that the assumed relation gives— ho ~ (IQ)^io > c = oi (®2;)^oi> c2o = (20)^20+ (102)%, hi= Ql)^io^oi > c21 = (21)&21 + (20 01)&2o&oi + (H 10)&11&10 + (102 01)610&oi. &c. ....•• and generally, in the expression of c^, every symmetric function of biweight pq of the quantities in I. occurs, each attached to the corresponding product of coefficients from the second identity. Now n(l + aAo£ + Mu’? + •• • + aVffbpqZPTfi + ...) is from II. equal to nn{i + a,af^+PsP{t v}, as can be seen by putting x=a£, y = psy. Hence, from the assumed relation and III., S log {1 + af£ + |3i2 77} = S4 S log {1 + a<a,/)| + PfiPy} ; s
and now, expanding and equating coefficients of £Pyq, (M)2=(^2)(M)i» a simple and important consequence of the assumed relation. It shows that the relation in question is unaltered by interchange of the quantities in I. and II. in such wise that as and a^11 and also ps and pf are transposed. Let the operations dpq, ~Dpq refer to I., and the corresponding operations of II. and III. be denoted by the same symbols with the addition of single and double dashes respectively. Write the assumed relation in the abbreviated form U M - a101W“2foWa8/33-" > then dpgU = (dpqUa1Pi)uausP2Ua-sp3' • • + • • • > and d d d d ( V t~ dbpq^wLbp+x, + 6 oitfhW+i when performed upon u^, gives PaQyP Q dpq^oxfis = asPs% V Uasfa’ hence d'pqU=(pq)fyqU)
and, replacing U by its expression, dncm = (pq), and in general dpqCrs — (pq)cr-p,s-q > dv„ causes every other single part function to vanish, and must cause any monomial function to vanisj^which does not comprise whence, regarding the coefficients bpq as functions of the coefficients Cpq only, one of the partitions of the biweight pq amongst its parts. Since dpq = "Zi(dpqCrs — (pq )dpq • p+g-^p + q — 1)! d dpg ) pq dsr In a precisely similar manner d^q = (pqdpq, and making use of the solutions of the partial differential equation ^ — 0 are the the established relation (pq)^ (pq)(pq)i, single bipart forms, omitting spq, and we have seen that the (pq)qdpq — (pq)idpq — (pq)dpq. solutions of D^^O are those monomial functions in which the part pq is absent. As a consequence of this, if we regard the assumed relation as One more relation is easily obtained, viz.: defining a transformation of the quantities in III. into either of r+5 d - = dpq - 7tio^p+l, - Aoi^.j+1 +... + (- ) A rfp+r, -H + • • • the sets of quantities occurring in I. and II., the operation g rl g da. (pq)qdpq is an invariant. We can now derive relations between Theory of Three Identities.—Let p q the operations Dyg, Dp2, D^ . For 1 + a10£c + a^y + ... + apqx y 4- .. (I-) = (1 + aix + /3i?/)(l + “2*p+9fay) ■ • • 1 + l + bioX + boiy+... + hpqx y + --’ ^10 + ^01 - 2(^20+ 2^n + ••• { ) = (1 + a^aj-b /3^h/)(l + a, fx + pq y)..- (H -) 1 + ciox +,2)c01y + ...2) + CpgXPy* + ... - £(To)c7" ho0 +1 ymd01 -{ 2 ?mdw + 2^(U)d"1 + y)dQ2} = (1 + a x + j3i y)(1 + a2x + ^2 V) • • • (IIL) and this leads to wherein x and y are to be regarded as undeterminal quantities, log (l + £T)10+iiD0i + ... + fyqT>pq+...) and the identities as merely expressing. relations between the coefficients on the left and the quantities a, ^ fj16 ” = S log (1 + ?as.Dio + t7/3sD01 + ... + fyqo-lpqs'Dpq +•••); Assume the coefficients and quantities in the identities I. and II. to be given, and the coefficients in III. to be then determined by and thus to 1+ D + + 1 + Cio£ + Coil? + ... + Cpq!;pT)7q + ...p g ^ io ^oi+ • • • + ••• = n(l + as&10£ + ft&oi^ + • • • + P t1>pgZ V +•••)» = n(l + ?asD" + 77&D" + ■ • • + fvYPsKl +-); S % and 77 being undetermined quantities.
