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 292 we find ( d d d YdT dY’ dZ’

ALGEBRAIC / d d d_ ^ — ( l> ^l> ^3J • • • ' da?’ dy dz dlt v Msi • •' "l, 2> "S-

Observe the notation, which is that introduced by Cayley into the theory of matrices which he himself created. . Just as cogrediency leads to a theory of covariants, so contragrediency leads to a theory of contra variants. If u, a quantic in x, y, z, be expressed in terms of new variables X, Y, Z ... ; and iff, y, be quantities contragredient to x, y, z, ... ) there are found to exist functions off, y, f, ..., and of the coefficients in u, which need, at most, be multiplied by powers of the modulus to be made equal to the same functions of 3, H, Z, ... of the transformed coefficients of u ; such functions are called contravariants of u. There also exist functions, which involve both sets of variables as well as the coefficients of u, possessing a like property ; such have been termed mixed concomitants, and they, like contravariants, may appertain as well to a system of forms as to a single form. As between the original and transformed quantic we have the umbral relations Aj = + X2&2 > -A-2 = A1!*! + Ai2*2 > and for a second form =Aj&j + X2&2 > B2—A ^2^2 • The original forms are a%, b™, and we may regard them either as different forms or as equivalent representations of the same form. In other words, B, b may be regarded as different or alternative symbols to A, a. In either case (AB) = A^a - AaBj = (y)(ab); and, from the definition, (ab) possesses the invariant property. We cannot, however, say that it is an invariant unless it is expressible in terms of the real coefficients. Since (a&) = a1&2-a2^i) that this may be the case each form must be linear ; and if the forms be different (ab) is an invariant (simultaneous) of the two forms, iffi. real expression being a^ - oq&Q. This will be recognized as the resultant of the two linear forms. If the two linear forms be identical, the umbral sets a2; blt b2 are alternative, are ultimately put equal to one another and (ab) vanishes. A single linear form has, in fact, no invariant. When either of the forms is of an order higher than the first (ab), as not being expressible in terms of the actual coefficients of the forms, is not an invariant and has no significance. Introducing now other sets of symbols C, D, ...; c, <#,... we may write (AB)*(AC);(BC)*... = ( y+J+*+-(aby(acy(bc)*..., so that the symbolic product (ab)i(acy(bc)k..., possesses the invariant property. If the forms be all linear and different the function is an invariant, viz., the thith power of that appertaining to ax and bx multiplied by the j power of that appertaining to ax and cx multiplied by &c. If any two of the linear forms, say px, qx, be supposed identical, any symbolic expression involving the factor (pq) is zero. Notice, therefore, that the symbolic product (ab)i(acy(bc)k... may be always viewed as a simultaneous invariant of a number of different linear forms bx, cx, ... . In order that (a&)*(ac)-;(6c)*... may be a simultaneous invariant of a number of different forms a”1, &”2, cj1, where n2, n3, ... may be the same or different, it is necessary that every product of umbrae which arises in the expansion of the symbolic product be of degree ni in ai, a% ; in the case of blt b2 of degree n2 ; in the case of Cj, c2 of degree n3; and so on. For these only will the symbolic product be replaceable by a linear function of products of real coefficients. Hence the condition is i+j + ...=n1) i + Jc + ...=n2) j+k + ...=nz, If the forms ^**■ be identical the symbols are alternative, and provided that the form does not vanish it denotes an invariant of the single form a™. There may be a number of forms ax,bx,c^,...a,nd wTe may suppose such identities between the symbols that on the whole only two, three, or more of the sets of umbrae are not equivalent; we will then obtain invariants of two, three, or more sets of binary forms. The symbolic expression of a covariant is equally simple, because we see at once that since Af, B|, Cj, ... are equal to ax, bx, cx, ...

FORMS

respectively, the linear forms ax, bx, cx, ... possess the invariant property, and we may write (AB/(AC)j(BC)fc...A|Bjq... = (Xii4)t+-?+fc+ - "(ab)l(ac)bc)k.. .apxblcTx..., and assert that the symbolic product (ab)x(ac)bcf... apbxcl... possesses the invariant property. It is always an invariant or covariant appertaining to a number of different linear forms, and as before it may vanish if two such linear forms be identical. In general itn will be simultaneous covariant of the different forms n a O'X ii Ox, Ccx s,... if Ll i+j+... + p=n1, ... + o' = n2, j + k + ... +t =n3, It will also be a covariant if the symbolic product be factorizable into portions each of which satisfies these conditions. If the forms be identical the sets of symbols are ultimately equated, and the form, provided it does not vanish, is a covariant of the „ form axn. The expression («5)4 properly appertains to a quartic ; for a quadratic it may also be w'ritten (ab)cdy, and would denote the square of the discriminant to a factories. For the quartic (ab)i=(ay)2-aj)^)i=ab - kalaffffl +Qaablb = «oa4 + 6a| - doqaj -(- a0ai = 2(a3ai - Joqag + 3a|), one of the well-known invariants of the quartic. For the cubic (abyaxbx is a covariant because each symbol a, b occurs three times ; we can first of all find its real expression as a simultaneous covariant of two cubics, and then, by supposing the two cubics to merge into identity, find the expression of the quadratic covariant, of the single cubic, commonly known as the Hessian. By simple multiplication (albffl - 2aa2bb2 + a^lb^xl + (abl - a1albb2 + (aa2bl -2a1a|616| +avffb^)x2 ; and transforming to the real form, (a3b2 - 2a1&1 + a2bQ)x + (a0b3 - aff2 - a2b1 + a3b3)x1x2 + (affz - 2a2b2 + a3b2)xl, the simultaneous covariant; and now, putting b = a, we obtain twice the Hessian (a0a2 - a)x + (a0a3- a1a2)x1x2 + (a^g - a?2)x?i. It will be shown later that all invariants, single or simultaneous, are expressible in terms of symbolic products. The degree of the covariant in the coefficients is equal to the number of different symbols a, b, c, ... that occur in the symbolic expression; the degree in the variables (i.e. the order of the covariant) is p +a + r +... and the weight * of the coefficient of the leading term xp+<T+T+ •" is equal to i+j+k+.... It will be apparent that there are four numbers associated with a co variant, viz., the orders of the quantic and covariant, and the degree and weight of the leading coefficient ; calling these n, e, d, w respectively we can see that they are not independent integers, but that they are invariably connected by a certain relation nd -2w = e. For, if <p(a3,.. .x^,x2) be a covariant of order e appertaining to a quantic of order n, ^(A0,.. .£i,£2) = (hy)w<p(a0,. + p-ffi) 5 we find that the left- and right-hand sides are of degrees nd and 2w + e respectively in Xx, /q, X2, /q, and thence nd = 2w + 2e. Symbolic Identities.—For the purpose of manipulating symbolic expressions it is necessary to be in possession of certain simple identities which connect certain symbolic products. From the three equations ax=alx1 +a2x2, bx = b1x1 + b^x2, cx=c1x1 + c2x2, we find by eliminating xx and x2 the relation aj.bc) + bx(ca) + cx(ab) = 0 . (I.) r ■ Introduce now new umbrae dx, d2 and recall that + d2, - dx, are cogredient with xx and x2. We may in any relation substitute for any pair of quantities any other cogredient pair so that + nkn.
 * The weight of a term a^a^...aff is defined as being ^ + 2^+ •••