* Proc. Camb. Phil. Soc., vol. xxiii, pp. 109–119.Google Scholar

† Atti della R. Accad. Lincei, vol. iii (1885–1886). I must thank Mr F. P. White (St John's College) for having drawn my attention to these memoirs.Google Scholar

* Cf. § 1 (1). Also Proc. London Math. Soc. 2, 24 (1925), p. 83.Google Scholar

* Called by Pittarelli (loc. cit.) “punti di diramazione.” He gives a few of their properties.Google Scholar

* I owe this symmetrical decomposition to a suggestion of Professor Baker, H. F..Google Scholar

* I omit brackets after operational symbols F, F⁻¹, to avoid confusion with functional notation.Google Scholar

* These are characteristic features of the automorphic linear group of a non-singular quadric in space of any odd number of dimensions, and are consequences of the existence of two algebraically distinct systems of generating regions on the quadric. In space of one dimension, the quadric is the point-pair, and the two points which constitute the pair answer to the two algebraically distinct systems of generating regions.Google Scholar

* Two pencils of cubics Γ, Γ′ are apolar pencils, if any member of either is apolar to any member of'the other, Γ is a null pencil, if any two of its members are apolar. If binary cubic forms are represented by points in three-space (as in sub-section 26 infra), the forms which are perfect cubes will correspond to a certain twisted cubic C. The null pencils will then correspond to the lines which belong to the tangent linear complex T of C; apolar pencils will correspond to polar lines of the complex T. It is well known that two polar lines of T are met by the same four tangents of C; the meaning of this is easily seen to be that apolar pencils have the same Jacobian.Google Scholar

* For properties of the syzygetic pencil of quartics, see Grace, and Young, , Algebra of Invariants, pp. 197–208Google Scholar. For the study of the syzygetic pencil in relation to pencils of cubics, Meyer's Apolarität may be consulted; Study, “On the irrational Covariants of certain Binary Forms,” Amer. Journ. of Math. 17 (1895) will also be found useful.Google Scholar

* Let p′ be one cross ratio of the point-pairs P ₁, P ₂, so that the other is . Then it is easily shown that the absolute invariant of (P₁, P₂) is , where . Again, if q′ be a cross ratio of (Px, Ps) and the pair of perfect squares in their pencil, and , the absolute invariant may be shown to be . If we represent quadratics as points in the plane of a fundamental circle of centre S and radius ρ, the points on the circle representing the perfect squares, then the absolute invariant of the quadratics represented by S and K may be shown to be, 1−μ² where SK=μp.Google Scholar

* 4·10 shows that the third absolute parameters of the singular pencil [P _l, P ₁], and the pencil containing a perfect cube, are 0 and 8 respectively. It will be an excellent verification of 4·10 to obtain these values directly. The former value can be obtained by the method of limits, the latter by considering the pencil

in respect of the representation [P ₁, P ₂], where

.

* This might be verified algebraically. The canonical form 1·2 of the auto-form of αβγ may be shown to be

where (βγ, ij) is short for (βi) (γj)+(βj)(γi).

Bv 3.6 the polar quadratic of this is

* For these two linear covariants and a related property of the Euler line, see Morley, F., “Some Polar Constructions,” Math. Ann. Bd. 51.Google Scholar