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ALGEBRAIC FORMS We may write therefore of causes such an invariant to vanish. Thus it has annihilators (W) r. ac u — h the forms CLX u -a — d + —a-,— d +- d +. k=- 1 0 d db1 db2 These forms, n in number, are called “ associated forms ” of/ (“ Schwesterformen ” “ formes associ^es ). d rph d 6r- d Every covariant is rationally expressible by means of the forms (%(Xq + l~j=~ dax + 2j=" da2 + • /, m-2, ms, ... mtc since, as we have seen, m0=;1, m1 = 0. It is easy and Gordan, in fact, takes the satisfaction of these conditions as to find the relations defining those invariants which Sylvester termed “ Combinants. ” «2 = |(/,/)2, The existence of such forms seems to have been brought to Sylvester’s notice by observation of the fact that the resultant of a*! and must be a factor of the resultant of a%. + nbvx and + for a common factor of the first pair must be also a «4 = |(/j/)4./2-|{(/J/)2}2, common factor of the second pair ; so that the condition for the and so on. existence of such common factor must be the same in the two To exhibit any covariant as a function of m0, ux, m2, take cases. A leading proposition states that, if an invariant of a^ cfy = + ady-fif1 and transform it by the substitution and fibx be considered as a form in the variables X and fi, and an f1Vl +/2l/2 = ? where /x = ~l,f= a2ax ~1, invariant of the latter be taken, the result will be a combinant of = f=f1X1+f2X2’ ax and bx. The idea can be generalized so as to have regard to thence *2^1 ~ ^2 ^ ternary and higher forms each of the same order and of the same f-Vi = +f2V ; f.y2=x£-fiV, number of variables. f.ay = axt + (af)r), For further information see Gordan. Vorlesungen iiber In- and variantentheorie, B. ii. § 6, Leipzig, 1887 ; E. B. Elliott. jM — 1 . an =mAjn, +(,2> ,, /n ^„n-3V3 +...+U ., yr, vto-2 V 2 +U> y 2£ S n Algebra of Quantics, Art. 264, Oxford, 1895. Associated Forms.—A. system of forms, such that every form Now a covariant of ax=f is obtained from the similar covariant appertaining to the binary form is expressible as a rational and integral function of the members of the system, is difficult to of by writing therein x1,x2, for yx, y2 and, since y1,y 2ha.ve obtain. If, however, we specify that all forms are to be rational, been linearly transformed to { and y, it is merely necessary to but not necessarily integral functions, a new system of forms form the covariants in respect of the form (m^ + m2??)" and then arises which is easily obtainable. A binary form of order n con- division, by the proper power off, gives the covariant in question tains n independent constants, three of which by linear trans- as a function of/,m0 = 1 ,m2,M3, ... un. formation can be given determinate values ; the remaining n-3 Fx. gr., in the case of the binary quartic, coefficients, together with the determinant of transformation, givens n-2 parameters, and in consequence one relation must z3. <4=£4 + 3AfV + 4^3 + (l/f* exist between any n-1 invariants of the form, and fixing upon n-2 invariants every other invariant is a rational function of its i > ^2 2 members. Similarly regarding as additional parameters, = |fW -A3-2^ and j = 6 «!, «2, 3 /

we see that every covariant is expressible as a rational function a2,a3, ai of n fixed covariants. We can so determine these n covariants that every other covariant is expressed in terms of them by a which is exactly the relation connecting the five ground forms. The above system /, m2,m3, ... m„ is not, however, .for many fraction whose denominator is a power of the binary form. system of associated forms. First observe that vith.fx=ax = lx =... ,/1 = c»1a^_1,/2 = a0a* _1, purposes the/ 2 most convenient Writing (/,/ ) * = G2a, ((/,/')2*,/") = H2*+i , Hermite and Clebsch + x we find fv 2' have shown that ms is expressible in terms of the s forms /, G2, H3,G4, ... Gs or Hs according as s is even or uneven. Hence, we may take this for an associated system appertaining to a form of fX and that thence every symbolic product is equal to a rational order s. G„ 2fc=W0) an-Zk-.n-Zk bx , x function of covariants in the form of a fraction whose denominator We have is a power of fx. Making the substitution in any symbolic pro—/ ac n-2k-l.n-2k n-1. xt H2fc+1 ax cx , duct the only determinant factors that present themselves in the numerator are of the form (a/), (6/), (c/),...and every symbol a but/. (ab) = (af)b (bf)a (acfil = (af) ; x x finally appears in the form k n k cLn -2k k „ Ak n 2k 2k

fk = {af) a x- . G2fc-/ =«* b:- {(-/)bx-(bf)axy n-%k- n-2k/ fc pk has / as a factor, and may be written f .uk for, observing that lhk+1 -r=ax—b1 nx~2k(af){(af)bx- (bf)ax V'o^/-=/• wo ; ^1 = 0=/.Mi; where m0=1 ,M1=:0, Whence expression, by the binomial theorem, gives assume that = (af)kax = ~ 2 u (*2kfJC ~ u21:fi ~ Cl)u2k - ul -f1 + ^2)u2k - 2u2f2 ~ 2kf2 > Taking the first polar with regard to y or, 2k-2 = 2 { u^Uq + (22:)M2fc _ 2^2 ~ (fiz)u2k - 3U3 (n - Jc)(af)kax ~k~lay + £(a/)fc~^“ab){n - l)b™~ G 2k-f kn 21c l l k n 2) 21 . = kin - 2)axv ~ ~ uy + nax ~ ayu } ~ ' , + . 1/ Jcf2k .k)ukf > and, writing/2 and —fl for yl and y2, and, similarly ,2k-2, N)t/2^ .2k -2 ._2k(2k-B)u _ U - ••■{Tr {n - £)(«/)fc+1% “ ~1 + - l){ab){af)k~ bf)anx ~ kbnx “ 2 k )uk+luk> H2fc+1-/ -«2&+1'm0 + 2! ~ 2k- 1 2 1 2k 1 =k{n-2)f.{uf)u ^- -. and, from these, we can express the members of the u system in terms of the G and H system. Moreover the second term on the left contains As the symbolic expression of forms with two series of cogredient variables, we take if k be uneven, and if k be even ; in either case the factor (af)bx-{bf)ax={ab)f, and therefore (n - 7#fc+1 + M ./= k{n - 2)f. (uf)ukn ~ 2fc “1; and fk+l ^ seen t° be of the form /. .

m-k k n-X X m-k k n-X X (kAxK «2ai a2xi y2' a a xm-fc k n-X X. ,, ,,the real, expression • i.being • 'Wton/w x A k Ax) k x i the form being of degree m in x^, x2 and of degree n in yx, y2, and the coefficients aka.k being arbitrary. It may happen that is the product of the two forms a™, a™, and then akaK=ak. aK ; this case is included in the general theory.