LAW OF RECIPROCITY 
H 2 of order 4 = 3 and degree 4 < m — 1, since H e is of degree 
^ 6. Finally, 
{H 2 ,f) 3 = {h x 2 h\ Cix 3 ) 3 = (ha) 2 {h'a)h' x = {h' 2 x , (ha) 2 a x ) = 0. 
Hence /, H, J, D form a fundamental system of covariants 
{cf. § 30). 
63. Higher Binary Forms. The concepts introduced by 
Gordan in his proof of the finiteness of the fundamental system 
of covariants of the binary p-ic enabled him to find * the 
system of 23 forms for the quintic, the system of 26 forms for 
the sextic, as well as to obtain in a few lines the system for 
the cubic (§ 62) and the quartic (§31). Fundamental sys 
tems for the binary forms of orders 7 and 8 have been deter 
mined by von Gall.f 
Gordan’s method yields a set of covariants in terms of 
which all of the covariants are expressible rationally and 
integrally, but does not show that a smaller set would not 
serve similarly. The method is supplemented by Cayley’s 
theory | of generating functions, which gives a lower limit 
to the number of covariants in a fundamental system. 
64. Hermite’s Law of Reciprocity. This law (§ 27) can be 
made self-evident by use of the symbolic notation. Let the 
form 
(j)=a x p = p x p = . . .=Oo(*l — PlX2)(xi— P2X2) . . . (#1 — PpXf) 
have a covariant of degree d, 
K =Uo d 2(pi— P2Hpi — P3) ; (p2— P3,) k ■ ■ • P1X2) 11 ■ ■ • (xi—ppX2) lp , 
so that each of the roots pi, . . . , p p occurs exactly d times 
in each product. Consider the binary d-ic 
f=a x d = b x d = . . . =co(*i — riXi) . . . (xi—raxo). 
* Gordan, Invariantentheorie, vol. 2 (1887), p. 236, p. 275. Cf. Grace and 
Young, Algebra of Invariants, 1903, p. 122, p. 128, p. 150. 
f Malhematische Annalen, vol. 17 (1880), vol. 31 (1888). 
X For an introduction to it, see Elliott, Algebra of Quantics, 1895, p. 165, p. 
247.