198 Mr TurnhuU, Some Geometrical Interpretations 



reciprocals. So also would be (ahcu) and (a^ya;), the latter being 

 of a type not arising for less than three quadrics. Though the 

 process by which Gordan arrived at such symbols as {A0iv) and 

 (a/3p) was purely analytic, it is interesting to observe that from 

 the geometrical point of view such analytical results were almost 

 inevitable. Below will be found several examples of the use of 

 this principle of duality. 



The fundamental forms. 



§ 4. A brief investigation would reveal the importance of the 

 following forms, to which special symbols are therefore attached. 



Let / denote a^-, f denote hx~, 



S „ u^", %' „ %^ 



n „ {Ap)\ W „ {Bpf, 



k „ {AjBxf, k' „ {Baxf, 



X " {Ahuf, X » {Bauf, 



77^2 „ (abpf, Dia „ (ct^pT, 



and C „ {AB)(Ap)(Bp). 



Some account of these forms may be found in Salmon, Analytical 

 Geometry of Three Dimensions (revised by Rogers), Yol. i, Ch. ix. 

 There %, %, %', S' are denoted by a, r, r', a' (§ 214) : A^, k' are the 

 T, r of § 215 ; n, 7ri2, O' are the ^, ^^, ^' of § 217. 



Invariants. 



§ 5. The irreducible invariants are a^, bj, (ABf, a^^, bj or 

 the A, 0, ^, @', A' of Salmon, § 200. In fact, there are no other 

 types, for two quadrics of any dimension n, than the n + 1 co- 

 efficients of X, in the discriminant of 



The five covariants and contravariants. 



I 6. The covariants (w + 1 in the case of w-ary forms*) are the 

 four quadrics /, k, k', f and the quartic J defined by 



apbaaj)x (AB) (A^x) (Bax). 



This is indeed the jacobian of the four quadrics, and represents 

 the four planes of the self-conjugate tetrahedron (c£ Salmon, § 233). 



* Of. Turnbull, ' Quadratics in n variables' (pp. 235-238), Camb. Phil. Trans., 

 Vol. XXI, No. viii. 



H 



