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and their focal conics; the lines a, a' are asymptotes to the focal hy-
perbola (p. 647), whatever the species of the surface may be refe-
rences (in Notes to pp. 648, 649) to the Lectures,* for the focal ellipse
of the Ellipsoid, and for several different generations of this last sur-

(d). General Exponential Transformation (p. 651), of the equation
of any central quadric;


p = xa+yVa2ß, (N1), with x2fa + y2f UVaa' = 1, (N1')

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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-

Focal Ellipses.

(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-

pendicular to this latter line (p. 649).

(h). Equation of Confocals (p. 652),

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ARTICLE 408.-On Circumscribed Quadric Cones; and on the

Umbilics of a central quadric,


* Lectures on Quaternions (by the present author), Dublin, Hodges and


(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;


Condition of Contact, of the same surface with the right line PP',

(ƒ (p, p')− 1)3 = (ƒp − 1) (ƒp' − 1) ;


this latter is also a form of the equation of the Cone, with vertex at
P. which is circumscribed to the same quadric (ƒp = 1).

(5). The condition (Q1) may also be thus transformed (p. 654),

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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;


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),

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this is shown (p. 664) to conduct, for central quadrics, to the first

P2D2TfUdp = h = const.;


where P is the perpendicular from the centre o on the tangent plane,



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and D is the (real or imaginary) semidiameter of the surface, which
is parallel to the tangent (dp) to the curve. The known equation
of Joachimstal, P.D const., is therefore proved anew; this last
constant, however, being by no means necessarily real, if the surface
be not an ellipsoid.


(b). Deduction (p. 665) of a theorem of M. Chasles, that the tan-
gents to a geodetic, on any one central quadric (e), touch also a common
confocal (e); and of an integral (p. 666) of the form,

ei sin2 v1 + €2 cos v1 = e, = const.,

which agrees with one of M. Liouville.


(c). Without the restriction (R1'), the differential of the scalar h
in (S1) may be thus decomposed into factors (p. 666),

dh = d. P-2D-2 = 2Svdvdp-1. Svdp-1d2p;



but, by the lately cited Section (III. iii. 5, p. 515), the differential
equation of the second order,

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with an arbitrary scalar variable, represents the geodetic lines on any
surface: the theorem (a) is therefore in this way reproduced.

(d). But we see, at the same time, by (Si"), that the quantity h,
or P.D=h, is constant, not only for the geodetics on a central quadric,
but also for a certain other set of curves, determined by the differen-
tial equation of the first order, Svdvdp = 0, which will be seen, in the
next Series, to represent the lines of curvature.

ARTICLE 410.-On Lines of Curvature generally; and in particu-
lar on such lines, for the case of a Central Quadric, .

(a). The differential equation (comp. 409, (d)),

Svdvdρ = 0,


represents (p. 667) the Lines of Curvature, upon an arbitrary surface;
because it is a limiting form of this other equation,

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which is the condition of intersection (or of parallelism), of the normals
drawn at the extremities of the two vectors p and p + Ap.

(b). The normal vector v, in the equation (Ti), may be multiplied
(pp. 673, 700) by any constant or variable scalar n, without any real
change in that equation; but in this whole theory, of the treatment
of Curvatures of Surfaces by Quaternions, it is advantageous to con-
sider the expression Svdo as denoting the exact differential of some
scalar function of p; for then (by pp. 486, 487) we shall have an equa-
tion of the form,

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which usually involves p also. For instance, we may write generally

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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,

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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),

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(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



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

following Series (411).

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-

pendicular to its own polar.

(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-
fore the locus of the point r' is usually a quartic curve, with p for a
double point, whereat two branches of the curve cut each other at right
angles, and touch the two lines of curvature.

(c). If the point P be one of a principal section of the given surface,
but not an umbilic, the cone (C) breaks up into a pair of planes, whereof
one, say (P), is the plane of the section, and the other, (P'), is perpen-
dicular thereto, and is not tangential to the surface; and thus the
quartic (b) breaks up into a pair of conics through P, whereof one is
the principal section itself, and the other is perpendicular to it.

(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

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