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 ALGEBRAIC

FORMS

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Symbolic Represeoitation of Symmetric Functions.—Denote the is transformed into the operator by the substitution aLS aS a.9 a a a (a0, a2, ...as, ...) —( o> ^o i> ^o 2> •••)> elementary symmetric function ky ^ 5 ’ • * ’ at pleasure ; so that the theory of the general operator is coincident with that then, taking oi equal to co, we may write of the particular operator dv For example, the theory of inl + a1x + apc‘i+...=(l+pix)(l+p3x)... = eaix = ea.2x=ea&:=... variants may be regarded as depending upon the consideration of the symmetric functions of the differences of the roots of the where equation a —A= s— / flPZ'-Ps — a0xn - (” )a1xn~1 + (” )aixn~2 - ... = 0 ; s! s! Further, let and such functions satisfy the differential equation m 1 + 1^ + bpt? +... + bmx ={ +  -f-... -t* TICtn—p ^n ~ 0. so that For such functions remain unaltered when each root receives the 1+ + atfr + ... = { + /,2 e 2 -b a2 2> 3> ••• become respectively a, a^ha^, a^ + ^h^, a3 + Sha2, ... and/(a0, a1} a2, a3,...) becomes 0 l+a1rb3pi+ ... + bmpn and hence the functions satisfy the differential equation. The <T P important result is that the theory of invariants is from a certain _ „<rial + a-2“2+ • ■ + a'ma.m point of view coincident with the theory of non-unitary symmetric functions. On the one hand we mayn staten that non-unitary by brackets and [ ] symmetric functions of the quansymmetric functions of the roots of a0x - axx ~^ + a^n~‘2‘ - ... = 0, Denote tities p and <r respectively. Then are symmetric functions of differences of the roots ot 1 + aqtl] + altl2] + «2[2] -b a?[l3] + aia2[21] -b a3[3] + ... a0aj”-l!(”)a1a3”-1 + 2!(”)a2a3n-2- ... = 0 ; + apfyfyz ■ ■ aprlP1P‘2Ps-' -Pm] + ’ • • and on the other hand that symmetric functions of the differ= 1 + &2(1) + 6?(12) + 62(2) + &?(13) + &A(21) + HZ) + ences of the roots of + bqibfbf. .Lq™(Hlmm~qm-1.-Hnqi) + ... - ("Vpe”-1 -b (” )a#Sn-‘2‘ - ... = 0, + cr2a2. • +  ... =0. then expressing the symmetricwefunctions of oq, cr2,_... H obtain by comparison with the An important notion in the theory of linear operators in middle series the symbolical representation of all symmetric functions in brackets appertaining to the quantities p^ p2, P3, — general is that of MaciVIalion s Tfiwl/t'il'LTicciT O'pcTCitoT ( . Theory of a Multilinear partial Differential Operator with Applications to To obtain particular theorems the quantities oq, oq, o-3,...crm are the Theories of Invariants and Reciprocants, ” Proc. Land. Math. auxiliaries which are at our entire disposal. Thus to obtain Soc., t. xviii. (1886), pp. 61-88). It is defined as having four Stroll’s theory of seoninvariants put elements, and is written bi = <rx + (t2+ ... +<rm=:[l] = 0 ; n) we then obtain the expression of non-unitary symmetric functions of the quantities p as functions of differences of the symbols am ^’2) * * • = 1 [^a0dan + {IJ- + V) (rn-lim 0 Ex. gr. bl^) with m-2 must be a term in m ! am-2Wi2l u J a 4 + Si'H(m-l)! 1 !, 0 ®2 + fZ. o 1! oi*0 (m-2) 2 ! ° ®1J a»+2 e<Tiai+<roa2 _ e<ri(ai ~ <*2) — ... -f icj-f (oj - Ct2) + ..., ,f m! .... m -1 m! m-2 + (/t+3y) and since bl=cr$ we must have (m-l)~!T!ao a3 + (m_2)!i!i!«0 ^2 m! m-3a s_9 + (22) = Kr(ai ~ “2)4 = " ^(“^2 + “i«D + ^“1 (m-3)!3!“° °/ an-b3 = 2a4 - 2aLa3 + al "]• as is well known. l- k fa . The operators Again, if oq, aq, (r3...<rm be the m, mth roots of —1, bx = b2—... the coefficient of a^a^a^... being = bm—x — 0 and bm=l, leading to «o3aI + «l3a2+— a0dai + 2alda2+- are seen to be (1, 0 ; 1,1) 1 + (m) -b (mr) + (m3) +... = ^iai+^+.. -b<~ and (1, 1 ; 1, 1) respectively. Also the operator of the Theory and a a m of Pure Reciprocants (see Sylvester Lectures on the New Theory .’. (m‘)=: of Reciprocants, Oxford, 1888) is ' ' ms!;(<r2 i-b<r2o2-b...-bam, m)* , (4, 1 ;2, l) = h4ao0C(1 + lO«oal3a2+6(2aOa2 + «l)3a3+-”} It will be noticed that (yu, v ; m, ri) = p{, 0 ; m, n) + i'(0, 1 ; m, n). The importance of the operator consists in the fact that taking any two operators of the system (yu, v ; m, n) ; (yu1, v1 ; m1, n1). the operator equivalent to (/4, v ; m, ri)(y.1, vl; on1, n1) - (/a1, id ; m1, n1)^, v ; m, n), known as the “alternant” of the two operators, is also an operator of the same system. We have the theorem {y.,v,m,ri)(y}, jdjm1, ti1) — (/d, v1; m1, nx){y., v ;m, «) = (^1, J'i! mi, nfj ; where 1 1 1 1 Ml = (m + m-l)|^-1(ya + «. i')-£(/x + ?i»' )j , 1 . .1 ,,1!' H m-1. m1——M*' -1 2 > 1 p )' v,1 = (n ' 1 - ■n.)i' ' mjon m1=on1 + m-l, n1 = n +n, and we conclude that qud “alternation” the operators of the system form a “group.” It is thus possible to study simultaneously all the theories which depend upon operations of the group.

and we see further that (oq^i + <72a2 + • ■ • +vanishes identically unless &=0 mod m. If m be infinite and 1 -b Zqa; + Zq*2 + ... = (1 +  we have the symbolic identity eo-2a2+0-2012+V3«3+- • • — eP101+P202+P303+- • •, and (oqcti + <r 2a2 + cr 303 + .. .)p - (piPi + p.fl2 + pA +. ■ .f. Instead of the above symbols we may use equivalent differential operators. Thus let ba = ufa0 + 2a + Safo^ + ... and let a, b, c, ... be equivalent quantities. Any function of differences of Sa, d^, dc, ... being formed the expansion being carried out, an operand a0 or b0 or c0... being taken and b, c,... being subsequently put equal to a, a non-unitary symmetric function will be produced. Ex. gr. (Sa - 5J)2(5(i - Sc) = (S2 - 28aSb + 52)(5„ - Sc) = 5* - 2d2a8b + 5Jl - 5% + 28a8b8c - 5% = 6a, - iaJ)x + 2a,bo - 2a,,c. + 2ai&2Ci_ 2b2cx = 2(al-3a1a2 + 3a?) = 2(3). The whole theory of these forms is consequently contained imI plicitly in the operation 5.