Writing these results Dif = D(1)f, D = D(2)f+D(l2)f, D3f = D(3)f+ (21)f+ D(13)f, s =1 (J1)11(J3)12(J3)13... j1!j2!j3!... where (J1) 11 (J2) 12 13. .. is a separation of (11 1 12213 3 ...) of specification (mM'8m"`28m"`3s...), placing s under the summation sign to denote the is Zs 3s specification involved, 141t412t!p31!...

1 a a a a d =aal+a laa2 a2aa3+... +an we may write in general D s f = ZD(p l p 2 p 3 ��) the summation being for every partition (piP2p3...) of s, and D(p iP2 p 3 ...)f being =2 (Dpifi)(DP2f2) (DL'h3f3)f4...f,n. Ex. gr. To operate with D2 upon (213) (214) (15), we have D (2)f = (13) (214) (15) + (213) (14) (15), D c1 2)f = (122) (213) (15) +(213) (213) (14) + (212) (214) (14), and hence D2f = (214) (15) (13) +(213) (15) (14) +(213) (212) (15) +(213)2(14) +(214) (212) (14).

Application to Symmetric Function Multiplication.-An example will explain this. Suppose we wish to find the coefficient of (52413) in the product (20(2' 4)(0). (15).

Write (213) (214) (15) =... +A(524) (13) +...; then D5D1D1 (213) (214) (15) =A; every other term disappearing by the fundamental property of D8. Since we have: D2D?(1 4)(1 4)(13) =A Dg34 (13)+2(14)(13)(12)} =A D 2 D3 12(1)()+7(13)(1)+2(14)()+6(13)(12)} =A D712(1)3=A.

where ultimately disappearing terms have been struck out. Finally A=6.12=72.

The operator d1= aoaai+aiaa2+a20a3+... 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 Ao,a0aa1 + ialaa2 + 2a2 a 3 +...

is transformed into the operator d 1 by the substitution (ac, al, a2, ���as, ���) _ (ao, Xoai, X 6 X i a 2, ���, XcX1..%s_las,���), 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 aox n - (i) a i x n - 1 + (z) a 2 x n 2 - ... = 0; and such functions satisfy the differential equation aoaa i +2a0a 2 +3a 2 aa 3 +... +na n _ i aa n = 0. For such functions remain unaltered when each root receives the same infinitesimal increment h; but writing x-h for x causes ao, a1, a 2 a3,... to become respectively ao, ai+hao, a2+2ha1, a 3 +3ha 2, ... and f(ae i a5, a 2, a3,...) becomes f+h(aoaai +2alaa2+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 sym metric functions of the roots of aox n - a l x n - 1 -{-a 2 x n - 2 - ... =o, are symmetric functions of differences of the roots of aox n - 1!(n)a4xn-1+2!()a2xn-2-... = 0; and on the other hand that symmetric functions of the differences of the roots of aox n (7)alxn-1+ (z)a2xn-2-... =0, are non-unitary symmetric functions of the roots of a xn-a l xn 1 a2 x n-2 -... = 0.

0 1! +2!

An important notion in the theory of linear operators in general is that of MacMahon's multilinear operator (" Theory of a Multilinear partial Differential Operator with Applications to the Theories of Invariants and Reciprocants," Proc. Lond. Math. Soc. t. xviii. (1886), pp. 61-88). It is definied as having four elements, and is written the coefficient of a0 o a1 a2 2 ... being !

mi, ! . The operators ko.ki.k2 aoaai+alaa2+��., a00a i +2a11, 2 +��� are seen to be (I, o; 1, I) and (I, I; I, I) 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) =2 4a 0 ea 1 +10acaiaa 2 +6(2aoaz+a 2 1) 0 9a3+... � It will be noticed that (�, v; m, n) =p(1, 0; m, n)+v(0, 1; m, n).

The importance of the operator consists in the fact that taking any two operators of the system

(I l, v; m, n); (Ill, v l : m l, n1),

the operator equivalent to 

(I l , v; m, n ) (111, v 1; ml, n1) - (i l l , v1; ml, n1) (/l, v; m, n), known as the " alternant " of the two operators, is also an operator of the same system. We have the theorem (I I, v; m, n) (/l l, v l; ml , n i) - (Il l, P 1; m l, n ') (/l, v; m, n) = (11, vl; ml, ni); where 1 /l1= (ml +m-1) ml (/l +nlv) - u-2 Cu '+nvl) 1 1 m-1 1 m1-1 vl =(n -n)vv-E ml / lY- m /lv, m i =7111+m-I, n1=nl+n,

and we conclude that qua " 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 8 s elementar symmetric function a s by al a 2 a3 ...at pleasure; then, Y y si, ,si,... p, taking n equal to 00, we may write 1 +aix +a2x2 +... _ (1 + p ix) (1 + P2x) ... = a l z = e a2z =e.3.=...

where s s a i a 2 a3 = =..

Further, let 1 -1-b i x+ b 2 x 2' +... +bmx m = (1 +Q 1 x) (1 +0 2 x)... (1 +umx); so that 1 +alal+a2a1 +... = (1 +Plat) (1 +P2(71)... = ePlal, 1 + a i Q 2+ a 2 0 2 +... _ (1 +PiQ2) (1 +P2(72)... =e2a2, 1 +aiam.+a2am+... = (1 +Plain) (1 = er,nam; and, by multiplication, II (1 +ala+a2a2+...) = II (1-}-biP+b2P 2 +... +bmP"`), a = e?l a' 1 °2 a 2 +.. +om a m .

Denote by brackets () and [ ] symmetric functions of the quantities p and a respectively. Then 1111 + a i[ 1 ]+ a i [ 12 1+a2[ 2 ]+ a 7 [ 13 ] +ala2[ 21 ]+a3[3]+-� + a p1 a p2 a P 3 �� .ap rn[Y1 p 2t' 3 ... i'mJ +-� . 1 + b l(1) + b (12) + b 2(2) +bi (13) + b 1b2(21) + b 3(3) +... +00 2 0 ..b qm (m qm m -1 qm-1 ...2 Q2 1 s1) -{-... 2 3 m = ealal+Q2a2.. +amam Expanding the right-hand side by the exponential theorem, and then expressing the symmetric functions of al, a2, ...a m, 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 p i, P2, P3,��� To obtain particular theorems the quantities a l, a 2, a 3 , ...a, n are auxiliaries which are at our entire disposal. Thus to obtain Stroh's theory of seminvariants put b1=0-1+a2+��.+0-m [1] =0; we then obtain the expression of non-unitary symmetric functions of the quantities p as functions of differences of the symbols a 2 , a2, a3, ...

Ex. gr. 14(22) with m =2 must be a term in eQial+?2a2= eri (a1-a2>=...-[-a1(a1-a2)4+... and since b2 =at we must have (22) =24(al-a2)4 = 24(a i+ a 2) -6(a? a2+ ala2)+4a2a2 =2a4-2ala3+a2 as is well known.

Again, if a i, a 2, a 3 ...a m , be the t " roots of -1, b 1 = b 2 =... = b n_1 =o

and b.= I, leading to 1 + (m) + (m 2) + (m 3) +... = ea lal+a2az+. .+omam (m8) =ms!(alai+a2a2+... +amaa.)sm, (ll, v; m P -"O a an + (l l + v) (ll +2 v) (m (11 +3v) +...], m - 2 2 !2 a 0 a 1 aan +2 ! 1 ! 1 ! a o -2 a1a2 ! 3 !a7-3ap ? aan +2 (m-1) ! 1 ! a0 a3 + (m-2) (m -m1)! 11 ! ao -ialaan+l m ! m _i m ! -1)!1 ! ao (m -2) m !