: and their focal conics; the lines a, a' are asymptotes to the focal hy- (d). General Exponential Transformation (p. 651), of the equation and p = xa+yVa2ß, (N1), with x2fa + y2f UVaa' = 1, (N1') this auxiliary vector ẞ is constant, for any one confocal (e); the expo- nent, t, in (N1), is an arbitrary or variable scalar; and the coefficients, x and y, are two other scalar variables, which are however connected with each other by the relation (N1′). (e). If any fixed value be assigned to t, the equation (N1) then re- presents the section made by a plane through a (p. 651), which sec- tion is an ellipse if the surface be an ellipsoid, but an hyperbola for either hyperboloid; and the cutting plane makes with the focal plane of a, a', or with the plane of the focal hyperbola, an angle = ¿tπ. (f). If, on the other hand, we allow t to vary, but assign to x and y any constant values consistent with (Ni), the equation (N1) then represents an ellipse (p. 651), whatever the species of the surface may be; z represents the distance of its centre from the centre o of the surface, measured along the focal line a; y is the radius of a right cylinder, with a for its axis, of which the ellipse is a section, or the radius of a circle in a plane perpendicular to a, into which that ellipse can be orthogonally projected: and the angle tπ is now the excentric anomaly. Such elliptic sections of a central quadric may be otherwise obtained from the unifocal form (c) of the equation of the surface; they are, in some points of view, almost as interesting as the known circular sections and it is proposed (p. 649) to call them Centro- (9). And it is obvious that, by interchanging the two focal lines a, a' in (d), a Second Exponential Transformation is obtained, with a Second System of centro-focal ellipses, whereof the proposed surface is the locus, as well as of the first system (ƒ), but which have their centres on the line a', and are projected into circles, on a plane per- ARTICLE 408.-On Circumscribed Quadric Cones; and on the 653-663 * Lectures on Quaternions (by the present author), Dublin, Hodges and CONTENTS. (a). Equations (p. 653) of Conjugate Points, and of Conjugate Diarts, with respect to the surface fp = 1, f(0, p)=1, (P1), and ƒ(p, p') = 0; (P1) Condition of Contact, of the same surface with the right line PP', (ƒ (p, p')− 1)3 = (ƒp − 1) (ƒp' − 1) ; (Q1) this latter is also a form of the equation of the Cone, with vertex at (5). The condition (Q1) may also be thus transformed (p. 654), (Q1') Fbeing a scalar function, connected with ƒ by certain relations of reciprocity (comp. p. 483); and a simple geometrical interpretation may be assigned, for this last equation. (e). The Reciprocal Cone, or Cone of Normals o at P' to the circumscribed cone (Qi) or (Q1'), may be represented (p. 655) by the very simple equation, F(o: Sp'o) = 1; (Q1′′) which likewise admits of an extremely simple interpretation. (4). A given right line (p. 656) is touched by two confocals, and other known results are easy consequences of the present analysis; for example (pp. 658, 659), the cone circumscribed to any surface of the system, from any point of either of the two real focal curves, is a time of revolution (real or imaginary): but a similar conclusion holds good, when the vertex is on the third (or imaginary) focal, and even more generally (p. 663), when that vertex is any point of the (known and imaginary) developable envelope of the confocal system. (e). A central quadric has in general Twelve Umbilics (p. 659), whereof only four (at most) can be real, and which are its intersections with the three focal curves: and these twelve points are ranged, three by three, on eight imaginary right lines (p. 662), which intersect the circle at infinity, and which it is proposd to call the Eight Umbilicar Generatrices of the surface. (f). These (imaginary) umbilicar generatrices of a quadric are found to possess several interesting properties, especially in relation to the lines of curvature: and their locus, for a confocal system, is a developable surface (p. 663), namely the known envelope (d) of that system. ARTICLE 409.-Geodetic Lines on Central Surfaces of the Second Order, (a). One form of the general differential equation of geodetics on an arbitrary surface being, by III. iii. 5 (p. 515), this is shown (p. 664) to conduct, for central quadrics, to the first P2D2TfUdp = h = const.; (S1) integral, xxix Pages. 664-667 and D is the (real or imaginary) semidiameter of the surface, which = (b). Deduction (p. 665) of a theorem of M. Chasles, that the tan- ei sin2 v1 + €2 cos v1 = e, = const., which agrees with one of M. Liouville. (S1') (c). Without the restriction (R1'), the differential of the scalar h dh = d. P-2D-2 = 2Svdvdp-1. Svdp-1d2p; - (S1") but, by the lately cited Section (III. iii. 5, p. 515), the differential with an arbitrary scalar variable, represents the geodetic lines on any (d). But we see, at the same time, by (Si"), that the quantity h, ARTICLE 410.-On Lines of Curvature generally; and in particu- (a). The differential equation (comp. 409, (d)), Svdvdρ = 0, (T1) represents (p. 667) the Lines of Curvature, upon an arbitrary surface; which is the condition of intersection (or of parallelism), of the normals (b). The normal vector v, in the equation (Ti), may be multiplied which usually involves p also. For instance, we may write generally Pages. 667-674 CONTENTS. the scalar g, and the vectors A, μ being real, and being generally* functions of p, but not involving do. (c). This being understood, the two† directions of the tangent dp, which satisfy at once the general equation (T1) of the lines of curvature, and the differential equation Svdp = 0 of the surface, are easily found to be represented by the two vector expressions (p. 669), UV vλ + UV vμ ; (T1") they are therefore generally rectangular to each other, as they have long been known to be. (d). The surface itself remaining still quite arbitrary, it is found useful to introduce the conception of an Auxiliary Surface of the Second Order (p. 670), of which the variable vector is p+p', and the equation is, or more generally const.; and it is proposed to call this surface, of which the centre is at the given point P, the Index Surface, partly because its diametral section, made by the tangent plane to the given surface at P, is a certain Index Curve (p. 668), which may be consi-, dered to coincide with the known "indicatrice" of Dupin. (e). The expressions (T1′′) show (p. 670), that whatever the given surface may be, the tangents to the lines of curvature bisect the angles formed by the traces of the two cyclic planes of the Index Surface (d), on the tangent plane to the given surface; these two tangents have also (as was seen by Dupin) the directions of the axes of the Index Curve (p. 668); and they are distinguished (as he likewise saw) from all other tangents to the given surface, at the given point P, by the condition that each is perpendicular to its own conjugate, with respect to that indicating curve: the equation of such conjugation, of two tangents and r', being in the present notation (see again p. 668), (ƒ). New proof (p. 669) of another theorem of Dupin, namely that if a developable be circumscribed to any surface, along any curve thereon, its generating lines are everywhere conjugate, as tangents to the surface, to the corresponding tangents to the curve. (9). Case of a central quadric; new proof (p. 671) of still another theorem of Dupin, namely that the curve of orthogonal intersection (p. 645), of two confocal surfaces, is a line of curvature on each. (h). The system of the eight umbilicar generatrices (408, (e)), of a central quadric, is the imaginary envelope of the lines of curvature on that surface (p. 671); and each such generatrix is itself an imaginary xxxi Pages. For the case of a central quadric, g, λ, μ are constants. + Generally two; but in some cases more. It will soon be seen, that three lines of curvature pass through an umbilic of a quadric. line of curvature thereon: so that through each of the twelve umbilics (see again 408, (e)) there pass three lines of curvature (comp. p. 677), whereof however only one, at most, can be real: namely two genera- trices, and a principal section of the surface. These last results, which are perhaps new, will be illustrated, and otherwise proved, in the ARTICLE 411.-Additional illustrations and confirmations of the foregoing theory, for the case of a Central* Quadric; and especially of the theorem respecting the Three Lines of Curvature through an Umbilic, whereof two are always imaginary and rectilinear, . (a). The general equation of condition (Ti), or SvAvAp = 0, for the intersection of two finitely distant normals, may be easily trans- formed for the case of a quadric, so as to express (p. 675), that when the normals at P and P' intersect (or are parallel), the chord PP′ is per- (b). Under the same conditions, if the point P be given, the locus of the chord PP' is usually (p. 676) a quadric cone, say (C); and there- (c). If the point P be one of a principal section of the given surface, (d). But if the given point P be an umbilic, the second plane (P′) becomes a tangent plane to the surface; and the second conic (e) breaks up, at the same time, into a pair of imaginary† right lines, namely the two umbilicar generatrices through P (pp. 676, 678, 679). (e). It follows that the normal PN at a real umbilic P (of an ellip- soid, or a double-sheeted hyperboloid) is not intersected by any other real normal, except those which are in the same principal section; but that this real normal PN is intersected, in an imaginary sense, by all the normals P'N', which are drawn at points p' of either of the two ima- ginary generatrices through the real umbilic P; so that each of these * Many, indeed most, of the results apply, without modification, to the case of the Paraboloids; and the rest can easily be adapted to this latter case, by the con- sideration of infinitely distant points. We shall therefore often, for conciseness, omit the term central, and simply speak of quadrics, or surfaces of the second It is well known that the single-sheeted hyperboloid, which (alone of central quadrics) has real generating lines, has at the same time no real umbilics |