Page:EB1911 - Volume 01.djvu/676

The two equations

${\frac {\partial f}{\partial \delta }}=5{\text{B}}\delta ^{4}+4{\text{B}}\delta ^{3}\rho )=0$,
${\frac {\partial f}{\partial \rho }}=5({\text{B}}\delta ^{4}-4{\text{A}}^{2}\rho ^{4})=0$,

yield by elimination of $\delta$ and $\rho$ the discriminant

${\text{D}}=64{\text{B}}-{\text{A}}^{2}$.

The general equation of degree 5 cannot be solved algebraically, but the roots can be expressed by means of elliptic modular functions. For an algebraic solution the invariants must fulfil certain conditions. When $R=0$, and neither of the expressions ${\text{A}}{\text{C}}-{\text{B}}^{2}$, $2{\text{A}}{\text{B}}-3{\text{C}}$ vanishes, the covariant $\alpha _{x}$ is a linear factor of $f$; but, when ${\text{R}}={\text{A}}{\text{C}}-{\text{B}}^{2}=2{\text{A}}{\text{B}}-3{\text{C}}=0$, $\alpha _{x}$ also vanishes, and then $f$ is a product of the form $j_{x}^{3}$ and of the Hessian of $j_{x}^{3}$. When $\alpha _{x}$ and the invariants ${\text{B}}$ and ${\text{C}}$ all vanish, either A or $j$ must vanish; in the former case $j$ is a perfect cube, its Hessian vanishing, and further $f$ contains $j$ as a factor; in the latter case, if $\rho _{x}$, $\sigma _{x}$ be the linear factors of $i$, $f$ can be expressed as $(\rho \sigma )^{5}f=c_{1}\rho _{x}^{5}+c_{2}\sigma _{x}^{5}$; if both ${\text{A}}$ and $j$ vanish $i$ also vanishes identically, and so also does $f$. If, however, the condition be the vanishing of $i$, $f$ contains a linear factor to the fourth power.

The Binary Sextic.—The complete system consists of 26 forms, of which the simplest are $f=a_{x}^{6}$; the Hessian ${\text{H}}=(ab)^{2}a_{x}^{4}b_{x}^{4}$; the quartic $i=(ab)^{4}a_{x}^{2}b_{x}^{2}$; the covariants $l=(ai)^{4}a_{x}^{2}$; ${\text{T}}=(ab)^{2}(cb)a_{x}^{4}b_{x}^{3}c_{x}^{5}$; and the invariants ${\text{A}}=(ab)^{6}$; ${\text{B}}=(ii')^{4}$. There are

5 invariants: $(a,b)^{6},~(i,i')^{4},~(l,l')^{2},~(f,l^{3})^{6},~((f,i),l^{4})^{8}$;
6 of order 2: $l,~(i,l)^{2},~(f,l^{2})^{4},~(i,l^{2})^{3},~(f,l^{3})^{5},~((f,i),l^{3})^{6}$;
5 of order 4: $i,~(f,l)^{2},~(i,l),~(f,l^{2})^{3},~((f,i),l^{2})^{4}$;
5 of order 6: $f,~p=(ai)^{2}a_{x}^{4}i_{x}^{2},~(f,l),~((f,i),l)^{2},~(p,l)$;
3 of order 8: ${\text{H}},~(f,i),~({\text{H}},l)$;
1 of order 10: $({\text{H}},i)$;
1 of order 12: ${\text{T}}$.

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. lxix. 323-354). August 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 success, 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 quadratics, 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 Sigmund 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 Eugenio 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 investigated by Perrin (S. M. F. Bull. xv. 45-61, 1887).

Ternary and Higher Forms.—The ternary form of order $n$ is represented symbolically by

$(a_{1}x_{1}+a_{2}x_{2}+a_{3}x_{3})^{n}=a_{x}^{n}$;

and, as usual, $b,c,d,\dots$are alternative symbols, so that

$a_{x}^{n}=b_{x}^{n}=c_{x}^{n}=d_{x}^{n}=\dots$.

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 $a_{x},~b_{x},~c_{x},\dots$ in such manner, that each of the symbols $a,~b,~c,\dots$ occurs $n$ times. Such a symbolic product, if its does not vanish identically, denotes an invariant or a covariant, according as factors $a_{x},~b_{x},~c_{x},\dots$ do not or do appear. To obtain the real form we multiply out, and, in the result, substitute for the products of symbols the real coefficients which they denote.

For example, take the ternary quadratic

$(a_{1}x_{1}+a_{2}x_{2}+a_{3}x_{3})^{2}=a_{x}^{2}$,

or in real form $ax_{1}^{3}+bx_{2}^{3}+cx_{3}^{3}+2fx_{2}x_{3}+2gx_{3}x_{1}+2hx_{1}x_{2}$. We can see that $(abc)a_{x}b_{x}c_{x}$ is not a covariant, because it vanishes identically, the interchange of $a$ and $b$ changing its sign instead of leaving it unchanged; but $(abc)^{2}$ is an invariant. If $a_{x}^{2}$, $b_{x}^{2}$, $c_{x}^{2}$ be different forms we obtain, after development of the squared determinant and conversion to the real form (employing single and double dashes to distinguish the real coefficients of $b_{x}^{2}$ and $c_{x}^{2}$),

$a(b'c''+b''c'-2f'f'')+b(c'a''+c''a'-2g'g'')$
$+c(a'b''+a''b'-2h'h'')+2f(g'h''+g''h'-a'f''-a''f')$
$+2g(h'f''+h''f'-b'g''-b''g')+2h(f'g''+f''g'-c'h''-c''h')$;

a simultaneous invariant of the three forms, and now suppressing the dashes we obtain

$6(abc+2fgh-af^{2}-bg^{2}-ch^{2})$,

the expression in brackets being the well-known invariant of $a_{x}^{2}$, 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 $ax^{3}+by^{3}+cz^{3}+6dxyz$ (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. $xy^{2}-4z^{3}+g_{2}x^{2}y+g_{3}x^{3}$, and also the special form $axz^{2}-4by^{3}$ of the cuspidal cubic. An investigation, by non-symbolic methods, is due to F. C. J. Mertens (Wien. Ber. xcv. 942-991, 1887). Hesse showed independently that the general ternary cubic can be reduced, by linear transformation, to the form

$x^{3}+y^{3}+z^{3}+6mxyz$,

a form which involves 9 independent constants, as should be the case; it must, however, be remarked that the counting of constants 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 independent 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 ${\text{S}}=(abc)(abd)(acd)(bcd)$ cf degree 4, which for the canonical form of Hesse is $m(1-m^{3})$; its vanishing indicates that the form is expressible as a sum of three cubes. The Hessian is symbolically $(abc)^{2}a_{x}b_{x}c_{x}={\text{H}}^{3}x$, and for the canonical form $(1+2m^{3})xyz-m^{2}(x^{3}+y^{3}+z^{3})$. By the process of Aronhold we can form the invariant ${\text{S}}$ for the cubic $a_{x}^{3}+\lambda {\text{H}}_{x}^{3}$, and then the coefficient of $\lambda$ is the second invariant ${\text{T}}$. Its symbolic expression, to a numerical factor près, is

$({\text{H}}bc)({\text{H}}bd)({\text{H}}cd)(bcd)$,

and it is clearly of degree 6.

One more covariant is requisite to make an algebraically complete set. This is of degree 8 in the coefficients, and degree 6 in the variables, and, for the canonical form, has the expression

$-9m^{6}(x^{3}+y^{3}+z^{3})^{2}-(2m+5m^{4}+20m^{7})(x^{3}+y^{3}+z^{3})xyz$
$-(15m^{2}+78m^{5}-12m^{8})x^{2}y^{2}z^{2}+(1+8m^{3})^{2}(y^{3}z^{3}+z^{3}x^{3}+x^{3}y^{3})$.

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 G. 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 Vorlesungen i. 288, also Math. Ann. xix. 529-552); Perrin (S. M. F. Bull. xviii. 1-80, 1890); Rosanes (Math. Ann. vi. 264); and Gerbaldi (Annali (2), xvii. 161-196).

Ciamberlini has found a system of 127 forms appertaining to three ternary quadratics (Batt. G. xxiv. 141-157).

A. R. 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 $x_{1}$, $x_{2}$, $x_{3}$, the point variables—and of $u_{1}$, $u_{2}$, $u_{3}$, the contragredient 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 ${\tfrac {1}{2}}(n+4)(n-1)$ 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 $n^{o}m^{ic}$ comprises

${\frac {1}{4}}(n+1)(n+2)(m+1)(m+2)-3$ individuals.

Bibliographical references to ternary forms are given by Forsyth (Amer. J. xii. p. 16) and by Cayley (Amer. J. iv., 1881). Clebsch, in 1872, in papers in Abh. d. K. Akad. d. U. zu Göttingen, t. xvii. and Math. Ann. t. v., established the important result that in the case of a form in $n$ variables, the concomitants of the form, or of a system of such forms, involve in the aggregate $n-l$ classes of