m! +(m -3) D 5(213) (214) (15) - (13) (14) (14), as= and and we see further that (alai +a2a2+...+amam) k vanishes identically unless (mod m). If m be infinite and 1 + b i x + b 2 x 2 +... (1 + a i x) (1+ = s i z we have the symbolic identity +02712+0.3x3+... ePl g l + P2P2 + P31 3 3 -f -.. �, and (alai +(72a2+a3a3+�� �) P = (Pith +P2t2 +P3f 3 3+ � � �) P � Instead of the above symbols we may use equivalent differential operators. Thus let =a10a0+2a20al+3a30a2+...
and let a, b, c, ... be equivalent quantities. Any function of differences of S a, S b, S c ,... 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-3b)2(Sa
Sc) = (Sa-23aSb +3b) (Oa - Se) =Sz - 23QSb+303 b - SQS c +23a3 b 3c - StSc = 6a 3 - 4a2b1 +2a,b 2 - 2a2c1 +2alblci - 2b2c1 =2 (al - 3a1a2+3a3) = 2 (3) .
The whole theory of these forms is consequently contained implicitly in the operation S. Symmetric Functions Several Systems 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 / al, a2, a3,...
132, 03,�.� we start with the fundamental relation (1+alx+aly)(1+a2x +t2Y) (1+a3x +03y)... = 1 +alox +aoly +a20x 2 +auxy +aG2y2 +... P y q +... As shown by L. Schlafli 1 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+Eaix+Esiy+ /al a2x 2 +Malt2xy -Z01023,2+��� The most general symmetric function to be considered is E 41 041 8424-3033..� .conveniently written in the symbolic form (pigi p2g2 p3go...)� Observe that the summation is in regard to the expressions obtained by permuting then suffixes I, 2, 3, ...n. The weight of the function is bipartite and consists of the two numbers Ep and Eq; 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 (l + a i x +01Y) (I + a 2x+l32Y) (1 + a3x+133y)... = 1 +(l A x +(01) y +(102) x2 +(1001)xy+(512)3,2 +(103)x 3 +(10201)x i y+(10 O12)xy2+ (013)y3+... where in general a pg = (10 P 010).
All symmetric functions are expressible in terms of the quantities ap g 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 ap q involves but a single unit.
The number of partitions of a biweight pq into exactly i biparts is given (after Euler) by the coefficient of a ,z xPy Q in the expansion of the generating function 1 - ax. 1 - ay. 1 - axe. 1 - 1aye. 1ax3.1ax2y. 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) =s� and take logarithms of both sides of the fundamental relation; we obtain slox +soot' = + (3ly) S20x 2 +2siixy+s02y 2 = E(aix+(3 ly) 2, &C., and siox+SOly - (S 20 x2 + 2s ii x y+ s ooy 2) +... log (1 +aiox +aol)/+...+apgxPyq+.... From this formula we obtain by elementary algebra 1) ! p, g 5
?
7r 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 s pg ,... in terms of elementary functions can be at once written down.
We can verify the relations s 30 -a310 -3a 20 a 10 + a30, S 21 - 02100 01 -a 2C a 01 -0 11 0 10 021 The formula actually gives the expression of q) by means of separations of (10P01'), which is one of the partitions of (pq). This is the true standpoint from which the theorem should be regarded. It is but a particular case of a general theory of expressibility.
To invert the formula we may write 1 +aiox+aoly+... +apgxPyq+... = exp {(siox+Solt') - s20 x 2+ 2siixy+S02y2)+���}, and thence derive the formula ? /,) (-)P+4-laP4 (p i+ g l - 1) ! C '" 1 S (p2 +q21)t ? 7r 2 (-)?,rl ,rl 7r2pl lql ')C)C p2 !g2 ! � �� 7r1! 72 !...s7114h sP242...
which expresses the elementary functions in terms of the single bipart functions. The similar theorem for n systems of quantities can be at once written down.
It will be � shown later that every rational integral symmetric function is similarly expressible.
The Function hpg. - As the definition of h pg we take 1 + nlox+naly+... +n,gxPyq+...
1 -(1aix - Rly) (1-a2x-R2y)...' and now expanding the (P1 right-hand side _ I ql)(P 2 +1721..Q1 /2172�..), h pg - pi p2 / J L' the summation being for all partitions of the biweight. Further writing 1 +hlox+holy+...+ hpgx P y {-...
1a i ox +... + (-) P+q a pg x P y +..., we find that the effect of changing the signs of both x and y is merely to interchange the symbols a and h; hence in any relation connecting the quantities pg with the quantities a pg we are at liberty to interchange the symbols a and h. By the exponential and multinomial theorems we obtain the results) 1,r -1 (E7r) ! Aal Ar2 7R1! 7 R 2L.�� P141 P242..� And In This A And H Are Interchangeable.
(pi+qi - 1 )! ('1 (p2+172-1)! 1,r 2 S�2 pi! qi! S ] l p2! q2!.�� S ...7f1! 7r2!...SPIQYP242..� Dif f erential Operations. - If, in the identity 1 (1 +anx = 1+aiox+aoly+a20x 2 +allxy+a02y 2 +..., we multiply each side by (I -�-P.x+vy), the right-hand side becomes 1 +(aio+1.1 ') x +(a ol+ v) y +...+(a p4+/ 1a P-1,4+ va Pr4-1) xPyq - - ...; hence any rational integral function of the coefficients an, say f (al ° , aol, ...) =f exp(�dlo+vdol)f d a P-1,4, dot = dapg
The rule over exp will serve to denote that i udio+ vdo h is to be raised to the various powers symbolically as in Taylor's theorem.
Writing
D = gi d od p! 1 exp(Adlo + vdol) = (1+/oD10+ v Doi +..�+ VQ +.�.)f;
now, since the introduction of the new quantities 1.1., v results in the addition to the function (plglp2g2p3g3...) of the new terms
A PI Pg1 (p 2q2 p 3g3���) +/ AP2Pg2 (p 1 g 1P343 ...)+/ Z3vg3 (p l g i p 2 g 2 ...)+ �,
we find
DP141(plqip2q2p3q3���) = (p 2 q 2 p 3 q 3���),
and thence
D P141 D P242 D P343 ��. (p g p ,g p ,g3 ���) = I;
while D rs f =o unless the part rs is involved in f. We may then state that D pg is an operation which obliterates one part pq when such part is present, but in the contrary case causes the function to)