D2D715 (13)(12)(12)+2(14)(12)(1)+2(13)(13)(1)1  =A D2D7112(12)(12)(1)+7(13)(1)(1)+2(14)(1)+6(13)(12)} =A D712(1)3 =A, where ultimately disappearing terms have been struck out . Finally A=6.12=72 . The operator dl=oat+a1aa2+a2aa3+... which is satisfied by every symmetric fraction whose partition contains no unit (called by Cayley non-unitary symmetric functions), is of particular importance in algebraic theories . This arises from the circumstance that the general operator Xaoaal + Xlalaa3 +X2a2aa3+•• is transformed into the operator d1 by the substitution ((lt, al, a2, ...ae....) = (ao, aoal, kX1a2, so that the theory of the general operator is coincident with that of the particular operator d1 . For example, the theory of invariants may be regarded as depending upon the consideration of the symmetric functions of the differences of the roots of the equation 00x" - (t)alxn-'+ (a)a2x"-2 - ... = 0 ; and such functions satisfy the differential equation aoaal +2a1aa.2 +3a,aa3 + ... +na"-1aa" = O . For such functions remain unaltered when each root receives the same infinitesimal increment h; but writing x—h for x causes a0, a1, a2 a3,... to become respectively ac, al+hac, a2+2ha1, a3+3ha2r ... and f(a;, a1, a:, a3,...) becomes .f+h(aoaal+2a,aa2+3a2aa3+...)f , and hence the functions satisfy the differential equation . The important result is that the theory of invariants is from a certain point of view coincident with the theory of non-unitary symmetric functions . On the one hand we may state that non-unitary symmetric functions of the roots of aox"—a,x"-'+a2x"-2—...=o, are symmetric functions of differences of the roots of tux"—1!(")chic' +2!(z)a2x"-2— ... =0; and on the other.hand that symmetric functions of the differences of the roots of a0x"-(a)a1x"-' +(a)a2x"Jt- ... =0, are non-unitary symmetric functions of the roots of a5x"--ti 2 ...

An important notion in the theory of linear operators in general is that of

MacMahon's multilinear operator (" Theory of a Multi-linear partial Differential Operator with Applications to the Theories of Invariants and Reciprocants," Proc . Land . Math . Soc. t. xviii . (1886), pp . 61-88) . It is defined as having four elements, and is written (µ, v; m, n) ! s,_1 =m -'aoaa" + (µ+v) (m 1) ! 1 !ao 0100"+l m i +(µ+2v) (in 1) ! 1 Sao -'a2+(m 2) !2!ao 201 (aa"+2 +(µ+3v) (m ! !1 !a9 s,_1a3 +(m -2) m 1 ! 1 !ass,-2a3a2 -1 ) m i +(m-3)!31a9 an 3 -3 a0 aan+t the coefficient of ajsai'a22.., being kalkmk2, . The operators ao0al+al8a2+..., aoaa1+2alaa2+... are seen to be (I, o; 1, 1) and (1, 1; 1, 1) respectively .

Also the operator of the Theory of Pure Reciprocants (see

Sylvester Lectures on the New Theory of Reciprocants, Oxford, 1888) is (4,1; 2, 1) = 4a0aa1+10acalaa2+6(2aoa.+a,2)0a3+ ... It will be noticed that (µ, v; m, n) =µ(1, 0; m, n) +v(0, 1; in, n) . The importance of the operator consists in the fact that taking any two operators of the system (µ, v; in, n) ; (µ', v1: m1, n1), the operator equivalent to (µ, v; m, n) (Ai, v1 ; m', n') - (Al, v1 ; m1, n') (A, v ; in, n) , known as the " alternant " of the two operators, is also an operator of the same system . We have the theorem // .... (tl, v; in, n) (µ1, v1 ; m', nl) - (µ', v1 ; m', n') (t1, v; in, n) = (Ail, vl ; m1, nl) ; where 1 µ1=(m'+m-1) m1(µ+n1v)-m(µ1+nv') m—1 m1—1 v1=(n'—n)v'v+ s,1 µ'v- m µv1 . ml=m'+m-1, ni = nl +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 . Symbolic Representation of Symmetric Functions.—Denote the , s elementary symmetric function a1 b Y a1 a2 a3 at Pleasure; then, S!S!,s—!' "' taking n equal to oo , we may write 1 +aix +a2x2 + ... _ (1 +Plx) (1 +mx) ... = eel . -= ea2z = ea3s = ... where a s e a.= al a2-a3-P1P2...Pa=S=s-S- Further, let 1 +blx+b2x2+...+bmx°' = (1 +alx) (1 +o2x) ... (1 +amx) ; so that 1 +al al+aur + ... = (1+Alai) (1+1)2a1) ...

=e01a1 1 +ala2+a2a2 + ... = (1 +Ala2) (1 +1)202) ... = e02°'", 1 +alas, +a2am+.•• _ (1 +pia.) (1 +piam) • .. = ePmam; and, by multiplication, 11(1 +(ha+a2a2+...) =11(1 +

hip+b2p2+ ... +bow'), a p = ealal+a2a2+..+amam Denote by brackets ( ) and [ ] symmetric functions of the quantities p and a respectively . Then 1 +al[l]+ai [121+a2[21+a 11 [131 +a1a2[211+a3[31+ ... +aplap2ap 3•.. a!' . [plp2p 3 • •. p,] + . 1 +bl(1) +b ? (12) +b2(2) +b7 (13) +b02(21) +b3(3) + ... +b41b22b33...b."(nz "m—1`, -1...2421`n)+ ... = ealal+a2a2..+am'M .

Expanding the right-hand

side by the exponential theorem, and then expressing the symmetric functions of al, 62, ...a., which arise, in terms of b1, b2, ••• b., we obtain by comparison with the middle series the symbolical representation of all symmetric functions in brackets ( ) appertaining to the quantities p1, p,, 1)s,••• To obtain particular theorems the quantities al, 0-2, a3, •••am are auxiliaries which are at our entire disposal . Thus to obtain Stroh's theory of seminvariants put b, =ai-Fax +•••+am = [11=0; we then obtain the expression of non-unitary symmetric functions of the quantities p as functions of differences of the symbols a1, a2, Ex. gr. ba (22) with m =2 must be a term in e01a1+02°2= e01(a1-02)= ... +4~a1 (al- a2)4+... and since ba =al we must have L (22)=24(a,-012)4=24(al+a2)-6(ala2 T 0laa)+4aiaa =2a4—2ala3+ail as is well known . Again, if al, 0-2, a3...a,,, be the m, mth roots of -1, b1=b2 = ... =b._1 =o and bm=1, leading to 1 + (m) + (m2) + (m3) + ... = ealar+02a2+..+amam (m3) =rns!(alal+a2a2+...+amam)°m, then and and we see further that (alai +(nas+...+amam)k vanishes identically unless k=o (mod m) . If m be infinite and 1+ bix +b2x2 + ... = (1 +aix) (1 +a2x) ... =es1: =ePsz _ ... , we have the symbolic identity e6lal+a2a2+a3a3+...=eP1R1+P2P2+P3Pa+..., and !~ R (alai+e2a2+a3a3+...)P = (Pit+P2132+PafRR 3a+,••)P.• Instead of the above symbols we may use equivalent differential operators . Thus let Sa=alaao+2a2aoi+3aada2+ ... and let a, b, c, ... be equivalent quantities .

Any function of differ- ences of &, bb, 5 being formed, the expansion being carried out, an operand ao or bo or co ... being taken and b, c,... being subsequently put equal to a, a non-unitary symmetric function will be produced . Ex. gr . (Sa-Sb)2(Sa—o) =(ba-2&&b+S6)(Sa-a,;) =61- 26:So +M,-So&c+2SaSb&c -S 6 Se = 6a3 - 4a2b1+2a1b2 - 2a2c1+2albicl -2b2c1 =2 (ai -3a1a2+3(13) =2(3) . The whole theory of these forms is consequently contained implicitly in the operation S . Symmetric Functions of Several Systems of Quantities.—It will suffice to consider two systems of quantities as the corresponding theory for three or more systems is obtainable by an obvious enlargement of the nomenclature and notation . Taking the systems of quantities to be a1, a2, as, ... t h, 132ppr 131••• we start with the fundamental relation (1+ aix + fliy) (1+ a2x + 02y) (1 +a3x+N3y) ... = 1 +alox+aoly+a2ox2 +aiixy+ao2y2 + ... +aD2xpy4+ ... As shown by L . Schlafli i this equation may be directly formed and exhibited as the resultant of two given equations, and an arbitrary linear non-homogeneous equation in two variables . The right-hand side may be also written 1 +Ea1x+ml3lY+Ea1a2x2+Zai$zxy += AY2+ ...

The most general symmetric function to be considered is Gael q41 aa2Y220.gaga ... conveniently written in the symbolic form (plgl p2g2 psgs...)• Observe that the summation is in regard to the expressions obtained by permuting then suffixes 1, 2, 3, ...n . The

weight of the function is bipartite and consists of the two numbers Ep and Eg; the symbolic expression of the symmetric function is a partition into biparts (multiparts) of the bipartite (multipartite) number Ep, Eq . Each part of the partition is a bipartite number, and in representing the partition it is convenient to indicate repetitions of parts by power symbols . In this notation the fundamental relation is written (1 +aix+$iy) (1 +a2x+R2y) (1 +a3x+R3YZ . =1+(1v)x+(01)y+(102)x2+(1U 01)xy+(012)y2 +(103)x3+(1-o20-1)x2y+(IT) (T2)xy2+(Ol3)Y3+ ... where in general a,,= (10P 014) . All symmetric functions are expressible in terms of the quantities aP4 in a rational integral form; from this property they are termed elementary functions; further they are said to be single-unitary since each part of the partition denoting aP4 involves but a single unit . The number of partitions of a biweight pq into exactly s biparts is given (after Euler) by the coefficient of aµxPy4 in the expansion of the generating function 1 1 — ax . 1 —ay . 1— ax2 . 1 — axy .

1 — ay2 . 1— ax3 . 1— ax2y . 1— axy2 . 1 — ay3 ... The partitions with one bipart correspond to the sums of

powers in the single system or unipartite theory ; they are readily expressed in terms of the elementary .functions . For write (pq) =SP4 and take logarithms of both sides of the fundamental relation; we obtain slox+soiy = E(aix+01Y) S2ox2+2511XY+SO2Y2 = (aix+Yly)2, &c., and Siox + SO1Y 1 (520x2 + 2Sri.xy +s02y2) + ... =log (1 +alox+aoly+ ... +aD4xPy4+...) . From this formula we obtain by elementary algebra ((p+q - 1) ! ~- 21) ! p q 5 (—) 7r1 2r2 !

... aPlgl ap2g2.' corresponding to

Thomas Waring's formula for the single system . The analogous formula appertaining to n systems of quantities which Vienna Transactions, t. iv . 1852: expresses sP4,... in terms of elementary functions can be at once written down . Ex. gr .