The Binary Sextic.-The complete system consists of 26 forms, of which the simplest are ; the Hessian ; the quartic ; the covariants , and the invariants A=(ab); B=(). There are 5 invariants: (a, b), (i, '), (1, 1'), (ƒ, 1³), ((f, i), 14); 6 of order 2:1, (i, 1), (ƒ, 12)4, (i, 1), (f, 13), ((f, i), 13)6; 5 of order 4: i, (f, 1)², (i, 1), (ƒ, 12)³, ((f, i), 12)4; 5 of order 6:f,p=(ai)³a, (f, 1), ((f, i), 1), (1, 1); 3 of order 8: H, (,), (H, 1); 1 of order 10: (H, ); 1 of order 12: T. For a further discussion of the binary sextic see Gordan loc. cit., Clebsch loc. cit. The complete systems of the quintic and sextic were first obtained by Gordan in 1868 (Journ. f. Math. Ixix. 323- 354). Von Gall in 1880 obtained the complete system of the binary octavic (Math. Ann. xvii. 31-52, 139-152, 456); and, in 1888, that of the binary septimic, which proved to be much more complicated (Math. Ann. xxxi. 318-336). Single binary forms of higher and finite order have not been studied with complete suc- cess, but the system of the binary form of infinite order has been completely determined by Sylvester, Cayley, MacMahon, and Stroh, each of whom contributed to the theory. As regards simultaneous binary forms, the system of two quad- ratics, and of any number of quadratics, is alluded to above, and has long been known. The system of the quadratic and cubic, consisting of 15 forms, and that of two cubics, consisting of 26 forms, were obtained by Salmon and Clebsch; that of the cubic and quartic we owe to Gundelfinger (Programm Stuttgart, 1869, 1-43); that of the quadratic and quintic to Winter (Programm Darmstadt, 1880); that of the quadratic and sextic to von Gall (Programm Lemgo, 1873); that of two quartics to Gordan (Math. Ann. ii. 227-281, 1870); and to Bertini (Batt. Giorn. xiv. 1-14, 1876; also Math. Ann., xi. 30-41, 1877). The system of four forms, of which two are linear and two quadratic, has been in- vestigated by Perrin (S. M. F. Bull. xv. 45-61, 1887). Ternary and Higher Forms.-The ternary form of order n is represented symbolically by n 7 (x+x+3)=ax; and, as usual, b, c, d,... are alternative symbols, so that a=br=c=d=.... To form an invariant or covariant we have merely to form a product of factors of two kinds, viz. determinant factors (abc), (abd), (bce), etc...., and other factors ar, br, Cr,... in such manner, that each of the symbols a, b, c,... occurs n times. Such a symbolic product, if it does not vanish identically, denotes an invariant or à covariant, according as factors a, b, c,... do not or do appear. To obtain the real form we multiply out, and, in the result, sub- stitute for the products of symbols the real coefficients which they denote. For example, take the ternary quadratic (x+x+x)²=1, 3 or in real form ax + bx + cx³+2fxx+2gxx+2hxx We can see that (abc) abc is not a covariant, because it vanishes iden- tically, the interchange of a and b changing its sign instead of leaving it unchanged; but (abe)2 is an invariant. If abbe different forms we obtain, after development of the squared deter- minant and conversion to the real form (employing single and double dashes to distinguish the real coefficients of band c²), a(b'e"+b"c-2f'f")+b(c'a" + d'a' 29'9") +c(a'b"+a"b-2h'h")+2f(gh" + g'h'-a'f"-a"f") +2g(h'f"+h"f-b'g'-b'g')+2h("d"+f"g-ch"-ch'); a simultaneous invariant of the three forms, and now suppressing the dashes we obtain 6(abc+2fgh-af²-bg²-ch²), the expression in brackets being the well-known invariant of c the vanishing of which expresses the condition that the form may break up into two linear factors, or, geometrically, that the conic may represent two right lines. The complete system consists of the form itself and this invariant. The ternary cubic has been investigated by Cayley, Aronhold, Hermite, Brioschi, and Gordan. The principal reference is to Gordan (Math. Ann. i. 90-128, 1869, and vi. 436-512, 1873). The complete covariant and contravariant system includes no fewer than 34 forms; from its complexity it is desirable to consider the cubic in a simple canonical form; that chosen by Cayley was
ax³+by+cz³+6dxyz (Amer. J. Math. iv. 1-16, 1881). Another form, associated with the theory of elliptic functions, has been considered by Dingeldey (Math. Ann. xxxi. 157-176, 1888), viz. xy2-423+goxy+93, and also the special form ax2-4by³ of the cuspidal cubic. An investigation, by non-symbolic methods, is due to Mertens (Wien. Ber. xcv. 942-991, 1887). Hesse showed independently that the general ternary cubic can be reduced, by linear transformation, to the form 3+ y³+23+6mxyz, a form which involves 9 independent constants, as should be the case; it must, however, be remarked that the counting of con- stants is not a sure guide to the existence of a conjectured canonical form. Thus the ternary quartic is not, in general, expressible as a sum of five 4th powers as the counting of constants might have led one to expect, a theorem due to Sylvester. Hesse's canonical form shows at once that there cannot be more than two independ- ent invariants; for if there were three we could, by elimination of the modulus of transformation, obtain two functions of the coefficients equal to functions of m, and thus, by elimination of m, obtain a relation between the coefficients, showing them not to be independent, which is contrary to the hypothesis. The simplest invariant is S=(abe)(abd)(acd) (bed) of degree 4, which for the canonical form of Hesse is m(1-m³); its vanish- ing indicates that the form is expressible as a sum of three cubes. The Hessian is symbolically (abc)2abc-H3, and for the canonical form (1+2m³)xyz - m²(x³+y³+z3). By the process of Aronhold we can form the invariant S for the cubic a+λH3, and then the coefficient of X is the second in variant T. Its symbolic expression, to a numerical factor près, is (Hhe)(Hbd)(Hed)(bcd), and it is clearly of degree 6. One more covariant is requisite to make an algebraically com- plete set. This is of degree 8 in the coefficients, and degree 6 in the variables, and, for the canonical form, has the expression -9m³ (x²+y+3)-(2m+5m+ 20m7)(x3+ y³+23)xyz -(15m2+78m-12nt³)ay+(1+8m³)(y33 +233 +x³y³). Passing on to the ternary quartic we find that the number of ground forms is apparently very great. Gordan (Math. Ann. xvii. 217-233), limiting himself to a particular case of the form, has determined 54 ground forms, and Maisano (Batt. G. xix. 198-237, 1881) has determined all up to and including the 5th degree in the coefficients. The system of two ternary quadratics consists of 20 forms; it has been investigated by Gordan (Clebsch-Lindemann's Vor- lesungen, i. 288, also Math. Ann. xix. 529-552); Perrin (S. M. F. Bull. xviii. 1-80, 1890); Rosanes (Math. Ann. vi. 264); and Ger- baldi (Annali (2), xvii. 161-196). Ciamberlini has found a system of 127 forms appertaining to three ternary quadratics (Batt. G. xxiv. 141-157). Forsyth has discussed the algebraically complete sets of ground forms of ternary and quaternary forms (see Amer. J. xii. 1-60, 115-160, and Camb. Phil. Trans. xiv. 409-466, 1889). He proves, by means of the six linear partial differential equations satisfied by the concomitants, that, if any concomitant be expanded in powers of x1, x2, xs, the point variables-and of u₁, ug, ug, the contra- gredient line variables-it is completely determinate if its leading coefficient be known. For the unipartite ternary quantic of order n he finds that the fundamental system contains (n+4)(n-1) individuals. He successfully considers the systems of two and three simultaneous ternary quadratics. In Part III. of the Memoir he discusses bi-ternary quantics, and in particular those which are lineo-linear, quadrato-linear, cubo-linear, quadrato-quadratic, cubo-cubic, and the system of two lineo-linear quantics. He shows that the system of the bi-ternary nome comprises (n+1)(n+2)(m+1)(m+2)-3 individuals. (n+1 Bibliographical references to ternary forms are given by Forsyth (Amer. J. xii. p. 16), and by Cayley (Amer. J. iv. 1881). IV. ENUMERATING GENERATING FUNCTIONS. Professor Michael Roberts (Quart. Math. J. iv.) was the first to remark that the study of covariants may be reduced to the study of their leading coefficients, and that from any relations connect- ing the latter are immediately derivable the relations connect- It has been shown above that a covariant, in ing the former. general, satisfies four partial differential equations. Two of these show that the leading coefficient of any covariant is an isobaric and homogeneous function of the coefficients of the form; the remaining two may be regarded as operators which cause the
