19. Def. Corresponding planes, which touch concyclics of the same species, such as ellipsoids, pass through straight lines (du-normals of the points of contact), which touch two concyclic hyperboloids of different species. These planes touch the developable formed by tangent planes common to the hyperboloids. The intercepts of corresponding planes on the axes are proportional to those axes. Their Boothian coordinates are thus related: The difference of the reciprocals of the squares of the perpendiculars from the centre on two corresponding planes is constant, and equal to the difference of the reciprocals of the squares of the axes. (The dual of Ivory's theorem). If OM, ON' denote the perpendiculars from the centre on the tangent planes to two concyclic conicoids, and OM', ON, those on the corresponding planes, MN' = M'N 20. Confocal conicoids are not co-tangential. Concyclic conicoids do not intersect each other. Since concyclic cones, which are asymptotic to concyclic conicoids, only meet in the vertex, there is no direct dual of MacCullagh's theorem on the intercept of the chord common to two focal conics. 21. The concluding theorems are connected with dual curvature. (1) If a developable be formed by two concyclics, and radii-vectores be drawn to the points of contact of any common tangent line, the intercept (P) of either radius-vector by a tangent plane parallel to the other radius is constant. (2) If a developable be formed by two concyclics, and r be the radius-vector drawn to a point in one of them, and p the perpendicular on a tangent plane, which is parallel to the central plane through the two above-named points of contact, rp is constant. T The dual theorem is important in the theory of geodesic lincs on conicoids. (3) The polar of a given point with respect to a system of concyclics passes through the du-normal of that concyclic on which the point lies. The polar of the point (f, g, h) with respect to a system of concyclics {~° (¦ ¦ + λ) + y2' ('{' + λ) + =3 (2 + 2) = 1} passes through the intersection of the two planes fx gy hz {te + 2 + b = 1, fx + gy + he = 0}, 2 b2 which determine the du-normal. (4) If three concyclics (A), (B), (C) are co-tangential, the directrices of the two focal conicoids of principal curvature of (4) for the common tangent plane are the polar planes of the point of contact with (4) with respect to (B) and (C). This is the dual of Dr. Salmon's theorem (p. 161). (5) The points of contact of the common tangent plane with (B) and (C) are the intersections of consecutive dutangent lines (or intersections of consecutive tangent planes) to the dual curves of curvature, which pass through the point of contact with (4). These dual curves of curvature may be thus defined. The vector-plane through the tangent to such a curve is perpendicular to the vector-plane through the du-tangent. The locus of the points of contact with (B) and (C) is the dual of the surface of centres of a conicoid, and may be called its du-centro-surface. Cheltenham, Fch. 4, 1871. from which X is to be eliminated: the second equation is here the derived function of the first in regard to λ, so that rationalling the first equation, the result is, as will be shown, of the form ',1,'=0, and the result is obtained by equating to zero the discriminant of the quartic function. Denoting for shortness the first equation by A+B+C+D=0, the rationalised form is (A* + B* + C1 + D* − 2A3B2 − 2 A2 C2 — 2 A2D2 --2BC-2B2D - 2 CD") -6443B2 C2D2 = 0, which is of the form where · (A + 2Bλ + Cλ2)2 + (a, b, c, d, eX1, λ)' =0, Writing I, J' for the two invariants we find without difficulty l' = I − } P + ▲3, J'=J− Q + }AP-A3, P = aC2 - 4b3C + 2c (AC + 2B2) — 4dAB + eA2, Q = (ce-d') A+ (ae + 2bd − 3c2). } (AC +242) + (ac − b2) C2 -2 (ad-be) Be The result thus is 3 (I-P+A) - 27 (J− Q+ JAPA3)” = 0, or what is the same thing, it is (1 − P)3 – 27 (J— Q)2 — 9AP (J – 2 Q) + ▲2(4Ï3−8IP+P2) +8A3 (J− 2 Q) + A+. 1®I = 0, = where the left-hand side is of the order 24 in (x, y, z, w). I apprehend that the order should be 12 only; for writing (x, y, z, w) in place of (x, y, z, w"), the equations which connect (a, b, c, d) express that these quantities are the coordinates of a point on a plane cubic; and the problem is in fact that of finding the reciprocal of the plane cubic: this is a sextic cone, or restoring (x2, y2, z2, w3) instead of (x, y, z, w), we should have a surface of the order 12. I cannot explain how the reduction is effected. TABLES OF THE BINARY CUBIC FORMS FOR THE DETERMINANTS. By Prof. CAYLEY. THE theory of binary cubic forms for determinants, as well positive as negative, has been studied by M. Arndt in the memoir "Versuch einer Theorie der homogenen Functionen des dritten Grades mit zwei Variabeln," Grunert's Archiv, t. XVII. (1851, pp. 1-54) and in the later memoir, "Tabellarische Berechnung der reducirten binären cubischen Formen und Klassification derselben für alle negativen Determinanten (-D) von D=3 bis D= 2000," ditto, t. XXXI. (1858), pp. 335-445, he has given a very valuable Table of the forms for a Negative Determinant. It has appeared to me suitable to arrange this Table in the manner made use of for Quadratic Forms in my memoir "Tables des formes quadratiques binaires pour les déterminants negatifs D=-1 jusqu'à D-100, pour les déterminants positifs non carrès depuis D2 jusqu'à D=99, et pour les treize determinants |