so we obtain P: = of the second pair; so that the condition for the existence of such common factor must be the same in the two cases. A leading proposition states that, if an invariant of Xax and i ubi be considered as a form in the variables X and ,u, and an invariant of the latter be taken, the result will be a combinant of cif and b1'. The idea_can be generalized so as to have regard to ternary and higher forms each of the same order and of the same number of variables.
For further information see Gordan, Vorlesungen Tiber Invariantentheorie, Bd. ii. � 6 (Leipzig, 1887); E. B. Elliott, Algebra of Quantics, Art. 264 (Oxford, 1895).
Associated Forms.-A system of forms, such that every form appertaining to the binary form is expressible as a rational and integral function of the members of the system, is difficult to obtain. If, however, we specify that all forms are to be rational, but not necessarily integral functions, a new system of forms arises which is easily obtainable. A binary form of order n contains n independent constants, three of which by linear transformation can be given determinate values; the remaining n-3 coefficients, together with the determinant of transformation, give us n -2 parameters, and in consequence one relation must exist between any n - I invariants of the form, and fixing upon n-2 invariants every other invariant is a rational function of its members. Similarly regarding 1 x 2 as additional parameters, we see that every covariant is expressible as a rational function of n fixed covariants. We can so determine these n covariants that every other covariant is expressed in terms of them by a fraction whose denominator is a power of the binary form.
First observe that with f x =a: = b z = ���,f1 = a l a z ', f 2 = a 2 az-', f x =f,x i +f 2 x i, we find (ab) - (a f) bx - (b f) ax. fx ? and that thence every symbolic product is equal to a rational function of covariants in the form of a fraction whose denominator is a power of f x. Making the substitution in any symbolic product the only determinant factors that present themselves in the numerator are of the form (af), (bf), (cf),...and every symbol a finally appears in the form.
% -k Y k = (af) k a n x. 'hc has f as a factor, and may be written f. uk; for observing that 1,to =f. =f. uo; 4, 1=0=f.; where u 0 =1, u1=o, assume that tfik = (af) k ay -k = f. u k =�y. ukx(n-2) � Taking the first polar with regard to y (n - k) (a f) xa x -k-l ay+ k (af) k-l ay -k (ab) (n -1) b12by n kn-2k-1 n-1 k(n-2) =k(n- 2)a u x u5+nax ayux and, writing f 2 and -f l for y1 and 3,21 (n-k)(a f) k+ta i k-1 + k (n - 1)(ab)(a f) k-1 (b f)4 1 k by-2 = (uf)u xn-2k-1? Moreover the second term on the left contains ( a f)' c -2b z 2 = 2 (a f) k-2b x 2 - (b) /0-2a 2 � if k be uneven, and (af)?'bx (i f) of) '-la if k be even; in either case the factor (af) bx - (bf) ax = (ab) f, and therefore (n-k),bk+1 +M�f = k(n-2)f.(uf)uxn-2k-1; and 4 ' +1 is seen to be of the form f .14+1. We may write therefore 1 These forms, n in number, are called " associated forms " of f (" Schwesterformen," " formes associbes ").
Every covariant is rationally expressible by means of the forms f, u 2, u3,... u n since, as we have seen uo =I, u 1 =o. It is easy to find the relations u2 =2(f u3 = ((f ,f')2,f") 114=2(f,f') 4 �f 2 41(1,f')212, and so on.
To exhibit any covariant as a function of uo, ul, a n = (aiy1+a2y2) n and transform it by the substitution fi y 1+f2 y where f l = aay 1 ,f2 = a2ay -1, x y - x y = X x thence f . y1 = x 15+f2n; f� y2 =x2-f?n, f .a b = ax+ (a f) n, l; n u 2 " 2 22 2 +` n) u3 n-3n3+...+U 2jn� 3 n Now a covariant of ax =f is obtained from the similar covariant of ab by writing therein x i, x 2, for yl, y2, and, since y?, Y2 have been linearly transformed to and n, it is merely necessary to form the covariants in respect of the form (u1E+u2n) n, and then division, by the proper power of f, gives the covariant in question as a function of f, u0 = I, u2, u3,...un.
Summary of Results.-We will now give a short account of the results to which the foregoing processes lead. Of any form az there exists a finite number of invariants and covariants, in terms of which all other covariants are rational and integral functions (cf. Gordan,, Bd. ii. � 21). This finite number of forms is said to constitute the complete system. Of two or more binary forms there are also complete systems containing a finite number of forms. There are also algebraic systems, as above mentioned, involving fewer covariants which are such that all other covariants are rationally expressible in terms of them; but these smaller systems do not possess the same mathematical interest as those first mentioned.
The Binary Quadratic.-The complete system consists of the form itself, ax, and the discriminant, which is the second transvectant of the form upon itself, viz.: (f, f') 2 = (ab) 2; or, in real coefficients, 2(a 0 a 2 a 2 1). The first transvectant, (f,f') 1 = (ab) a x b x ,vanishes identically. Calling the discriminate D, the solution of the quadratic as =o is given by the formula a: = o ( a0+a12_x2 (a0x+aix2 If the form a 2 be written as the product of its linear factors p.a., the discriminant takes the form -2(pq) 2. The vanishing of this invariant is the condition for equal roots. The simultaneous system of two quadratic forms ai, ay, say f and 0, consists of six forms, viz.
the two quadratic forms f, 4); the two discriminants (f, f')2,(0,4')2, and the first and second transvectants of f upon 4, (f, ,>) 1 and (f, 402, which may be written (aa)a x a x and (aa) 2 . These fundamental or ground forms are connected by the relation - 2 1 (f,4) 1) 2 = -2f4,(f ,4,)2+ 02(f,f')2.
If the covariant (f,4) 1 vanishes f and 4 are clearly proportional, and if the second transvectant of (f, 4 5) 1 upon itself vanishes, f and 4) possess a common linear factor; and the condition is both necessary and sufficient. In this case (f, �) 1 is a perfect square, since its discriminant vanishes. If (f,4) 1 be not a perfect square, and rx, s x be its linear factors, it is possible to express f and 4, in the canonical forms Xi(rx)2+X2(sx)2, 111(rx)2+1.2 (sx) 2 respectively. In fact, if f and 4, have these forms, it is easy to verify that (f, 4,)i= (A j z) (rs)r x s x . The fundamental system connected with n quadratic forms consists of (i.) the n forms themselves f i, f2,�� fn, (ii.) the (2) functional determinants (f i ,f k) 1 , (iii.) the (n 2 1) in variants (f l, fk) 2, (iv.) the (3) forms (f i, (f k, f ni)) 2 , each such form remaining unaltered for any permutations of i, k, m. Between these forms various relations exist (cf. Gordan, � 134).
The Binary Cubic.-The complete system consists of f=aa,(f,f')'=(ab)2a b =0 2 ,(f 0)= (ab) 2 (ca)b c=Q3, x x x x x x and (0,0')2 (ab) 2 (cd) 2 (ad) (bc) = R.
To prove that this system is complete we have to consider (f, o) 2, 04') 1, (f,Q) 1, (f,Q) 2, (f,Q) 3, 0,Q) 1, (o,Q)2, and each of these can be shown either to be zero or to be a rational integral function of f, 0 Q and R. These forms are connected by the relation 2Q2+ 3+Rf2=0.
The discriminant of f is equal to the discriminant of 0, and is therefore (0, 0') 2 = R; if it vanishes both f and 0 have two roots equal, 0 is a rational factor of f and Q is a perfect cube; the cube root being equal, to a numerical factor pres, to the square root of A. The Hessian 0 =A 2 is such that (f, 2 and if f is expressible in the form X(p x) 3 +,i(g x) 3 , that is as the sum of two perfect cubes,. we find that Di must be equal to p x g x for then t x (p x) 3 +, u (g x) 3, Hence, if px, qx be the linear factors of the Hessian 64, the cubic can be put into the form A(p x) 3 +�(g x) 3 and immediately solved. This method of solution fails when the discriminant R vanishes, for then the Hessian has equal roots, as also the cubic f. The Hessian in that case is a factor of f, and Q is the third power of u2,...
