From these formulae we derive two important relations, dp4 = or the function F, on the right which multiplies r, is said to be a simultaneous invariant or covariant of the system of quantics. This notion is fundamental in the present theory because we will find that one of the most valuable artifices for finding invariants of a single quantic is first to find simultaneous invariants of several different quantics, and subsequently to make all the quantics identical. Moreover, instead of having one pair of variables x i, x2 we may have several pairs yl, y2; z i, z2;... in addition, and transform each pair to a new pair by substitutions, having the same coefficients a ll, a12, a 21, a 22 and arrive at functions of the original coefficients and variables (of one or more quantics) which possess the above definied invariant property. A particular quantic of the system may be of the same or different degrees in the pairs of variables which it involves, and these degrees may vary from quantic to quantic of the system. Such quantics have been termed by Cayley multipartite.

Symbolic Form.-Restricting consideration, for the present, to binary forms in a single pair of variables, we must introduce the symbolic form of Aronhold, Clebsch and Gordan; they write the form

Iln n n-1 n-1 n n n aixi+a2x2) = 44+(1) a l a 2 x 1 x2+...+a2.x2=az

wherein al, a2 are umbrae, such that

n-1 n-1 n a 1, a 1 a 2 ,...a 1 a 2 , a2

are symbolical respreentations of the real coefficients �o, ai,... an_1 i a n, and in general a n-k a 2 is the symbol for Q k. If we restrict ourselves to this set of symbols we can uniquely pass from a product of real coefficients to the symbolic representations of such product, but we cannot, uniquely, from the symbols recover the real form, This is clear because we can write n-1 n-2 2 2n-3 3 a1a2 =a l a 2, a 1 a 2 = a 1 a2 while the same product of umbrae arises from n n-3 3 2n-3 3 aoa 3 = a l .a a 2 = a a 2 .

Hence it becomes necessary to have more than one set of umbrae, so that we may have more than one symbolical representation of the same real coefficients. We consider the quantic to have any n number of equivalent representations a- b n -c n So that a 1 -k a 2 = b l -k b 2 - c 1 -k c 2 = ... = a k; and if we wish to denote, by umbrae, a product of coefficients of degree s we employ s sets of umbrae.

n-1 2 Ex. gr. We write;L 22 = a 1 a 2 .b 1 n-2 b2s 3 n - 3 3 n-3 3 n-3 3 a 3 = a 1 a 2 .b 1 b 2 .c 1 c2, and so on whenever we require to represent a product of real coefficients symbolically; we then have a one-to-one correspondence between the products of real coefficients and their symbolic forms. If we have a function of degree s in the coefficients, we may select any s sets of umbrae for use, and having made a selection we may when only one quantic is under consideration at any time permute the sets of umbrae in any manner without altering the real significance of the symbolism.

Ex. gr. To express the function aoa2 - _ which is the discriminant of the binary quadratic aoxi -+-2a1x2x2-+a2x2 = ai =1, 1, in a symbolic form we have 2(aoa 2 -ai) =aoa2 +aGa2 -2 a1 � al = a;b4 -}-alb? -2ala2blb2 = (aib2-a2b1)2.

Such an expression as

a l b 2 -a 2 b i, which is aa 2 ab 2 aa x 2 2 ax1'

is usually written (ab) for brevity; in the same notation the determinant, whose rows are a l, a 2, a3; b2, b 2, b 3; c 1, c 2, c 3 respectively, is written (abc) and so on. It should be noticed that the real function denoted by (ab) 2 is not the square of a real function denoted by (ab). For a single quantic of the first order (ab) is the symbol of a function of the coefficients which vanishes identically; thus

(ab) =a1b2-a2bl= aw l -a1ao=0

and, indeed, from a remark made above we see that (ab) remains unchanged by interchange of a and b; but (ab), = -(ba), and these two facts necessitate (ab) = o.

To find the effect of linear transformation on the symbolic form of quantic we will disuse the coefficients a 111 a 12, a21, a22, and employ A1, I�1, A2, �2. For the substitution

rr xl =A 11 +1 2 12, 52=A21+�2E2, of modulus A1 �i = (Al�.2-A2�1) = (AM), A 2 �2

the quadratic form a k xi -1-2a 1 x i x 2 +a 2 4 = x =f (x), becomes

A41 +2A1E16 =At = OW, where Ao = aoA i +2a1AiA2 +a2Az, _ _ A 1 = ao A l�l +ai(A1/.22+A2�1) +7,2X2/22, A2 = ao�l +2a1�1/�2 +a 2�2 �

We pass to the symbolic forms

a:= (aixi+a2x2) 2, A 2 = (A 151+ A 26) 2/

by writing for ao, al, a2 the symbols ai, a 1 a 2, a? A 1, A2 � Ai, A 1 A 2, A2 and then Ao = al Ai+2a1a2AIA2+a2 A2 - (a1A1+a2A2) 2 = a?, A l = (a 1 A 1 +a2A2) (al�l +a2�2) = aAa�, A 2 = (al�l +a2/-12) 2 = aM; so that A = aa l +2a A a u 152+aM5 2 = (aA6+a,e2)2; whence A1, A 2 become a A, a m, respectively and ?(S) = (a21+a,E2) 2;

The practical result of the transformation is to change the umbrae a l , a 2 into the umbrae a s = a1A1 +a2A2, a � = a1/�1 + a21=2 respectively.

By similarly transforming the binary n ic form ay we find

Ao = (aI A 1 +a2 A2) n = aAn A l = (alAi - I -a 2 A 2) n1 (a1�1 +a2m2) = aa a � - A i n-1 A2, n-k k n-k k n-k k A = (al l+a2A2) (al�1+a2�2) = a A � =A 1 A2, so that the umbrae A1, A 2 are a A, a � respectively.

Theorem.-When the binary form

a y = (alxl +a2x2)n is transformed to A;. = (A11+A22)n

by the substitutions 51 = A l, E1+�1 2, 52 = A2E1+�2E2, the umbrae Al, A2 are expressed in terms of the umbrae al, a 2 by the formulae A l = Alai +A2a2, A2 = �la1 +�2a2�

We gather that A1, A2 are transformed to a l, a 2 in such wise that the determinant of transformation reads by rows as the original determinant reads by columns, and that the modulus of the transformation is, as before, (A / .c). For this reason the umbrae A1, A 2 are said to be contragredient to xi, x 2. If we solve the equations connecting the original and transformed unbrae we find

(A �) (- a 2) =A i( - A 2) + �'1A1,

(A �) a1 = A2(- A2)+�2A1,

and we find that, except for the factor (A /), -a 2 and +ai are trans formed to -A 2 and +A i by the same substitutions as x i and x 2 are transformed to i and E2. For this reason the umbrae -a 2, a l are said to be cogredient to 5 1 and x 2. We frequently meet with cogredient and contragedient quantities, and we have in general the following definitions:-(i) " If two equally numerous sets of quantities x, y, z,... x', y', z',... are such that whenever one set x, y, z,... is expressed in terms of new quantities X, Y, Z, ... the second set x', y', z', ... is expressed in terms of other new quantities X', Y', Z', .... by the same scheme of linear substitution the two sets are said to be cogredient quantities." (2) " Two sets of quantities x, y, z, ...; E, n, i, ... are said to be contragredient when the linear substitutions for the first set are

x =A1X+u1Y-}-v1Z-?--...,

y = A2X+,u2Y +v2Z �...,

Z = A 3 X +�3Y -1v 3 Z - -...,

and these are associated with the following formulae appertaining to the second set,

X = A1?+A277+A3? +...,

H =/.G1rr+�27]+�3? + ���,

Z = v16+v2%/+v3" +���,

wherein it should be noticed that new quantities are expressed in terms of the old, as regards the latter set, and not vice versa."

Ex. gr. The symbols - dy, d z, ... are contragredient with the d- variables x, y, z, ... for when

( x , z, ���) = (A l, �i, VI I ���)

(X, Y, Z, ���), I A 2, / 2 2, Y2, ... I I A S, 1 2 3, Y 3, .... 1

(Tr (T d d d d d d ,.. rd Y' ' ...) = 01, A2, A 3, ...)

(d ' ' z / 2 1, /22, / 1 3, ... Pl, P2, P3, ...