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℗ being the vector of the vertex P, and p + Ap that of any other point
P' of the cone; while a, a' are still, as in 407, (a), two real focal lines,
of which the lengths are here arbitrary, but of which the directions
are constant, as before, for a whole confocal system.
(A). This cone (C), or (V1), is also the locus (p. 678) of a system
* It might be natural to suppose, from the known general theory (410, (c))
of the two rectangular directions, that each such generatrix PP' is crossed perpendi-
cularly, at every one of its non-umbilicar points r', by a second (and distinct,
although imaginary) line of curvature. But it is an almost equally well known
and received result of modern geometry, paradoxical as it must at first appear, that
when a right line is directed to the circle at infinity, as (by 408, (e)) the gene-
ratrices in question are, then this imaginary line is everywhere perpendicular to
itself. Compare the Notes to pages 459, 672. Quaternions are not at all re-
sponsible for the introduction of this principle into geometry, but they recognise
and employ it, under the following very simple form: that if a non-evanescent
vector be directed to the circle at infinity, it is an imaginary value of the symbol 01
(comp. pp. 300, 459, 662, 671, 672); and conversely, that when this last symbol
represents a vector which is not null, the vector thus denoted is an imaginary line,
which cuts that circle. It may be noted here, that such is the case with the reci-
procal polar of every chord of a quadric, connecting any two umbilics which are not
in one principal plane'; and that thus the quadratic equation (XXI., in p. 669)
from which the two directions (410, (c)) can usually be derived, becomes an iden-
tity for every umbilic, real or imaginary: as it ought to do, for consistency with
the foregoing theory of the three lines through that umbilic. And as an addi-
tional illustration of the coincidence of directions of the lines of curvature at any
non-umbilicar point p' of an umbilicar generatrix, it may be added that the cone
of chords (0), in 411, (b), is found to touch the quadric along that generatrix,
when its vertex is at any such point p'.
of three rectangular lines; and if it be cut by any plane perpendicular
to a side, and not passing through the vertex, the section is an equila-
(i). The same cone (C) has, for three of its sides PP', the normals
(p. 677) to the three confocals (p. 644) of a given system which pass
through its vertex P; and therefore also, by 410, (g), the tangents
to the three lines of curvature through that point, which are the inter-
sections of those three confocals.
(j). And because its equation (V1) does not involve the constant
l, of 407, (a), (b), we arrive at the following theorem (p. 678):-Iƒ
indefinitely many quadrics, with a common centre o, have their asymp
totic cones biconfocal, and pass through a common point P, their normals
at that point have a quadric cone (C) for their locus.
ARTICLE 412.-On Centres of Curvature of Surfaces,
the equation (Wi′′) being a new form of the general differential equa-
tion of the lines of curvature.
(b). Deduction (pp. 680, 681, &c.) of some known theorems from
these equations; and of some which introduce the new and general
conception of the Index Surface (410, (d)), as well as that of the
known Index Curve.
(c). Introducing the auxiliary scalar (p. 682),
in which g, λ, μ are, for any given point P, the constants in the equa-
tion (U1") of the index surface; the difference of the two curvatures
R therefore vanishes at an umbilic of the given surface, whatever the
form of that surface may be: that is, at a point, where v || or || μ,
and where consequently the index curve is a circle.
(d). At any other point P of the given surface, which is as yet en-
tirely arbitrary, the values of r may be thus expressed (p. 681),
a1, a2 being the scalar semiaxes (real or imaginary) of the index curve
(defined, comp. 410, (d), by the equations Sp'op' = 1, Svp' = 0).
(e). The quadratic equation, of which r1 and r1⁄2, or the inverse
squares of the two last semiaxes, are the roots, may be written (p. 683)
under the symbolical form,
which may be developed (same page) into this other form,
the linear and vector functions,
and x, being derived from the func-
= Udp, r' = U (vdp), (A2), and therefore rr'= Uv, (A2)
are unit tangents to the lines of curvature, it is easily
this general parallelism of dr' to r being geometrically explained, by
observing that a line of curvature on any surface is, at the same time,
a line of curvature on the developable normal surface, which rests upon
that line, and to which r' or vr is normal, if r be tangential to the
(h). If the vector of curvature (389) of a line of curvature be
projected on the normal v to the given surface, the projection
(p. 686) is the vector of curvature of the normal section of that sur-
face, which has the same tangent r; but this result, and an analo-
gous one (same page) for the developable normal surface (g), are
virtually included in Meusnier's theorem, which will be proved by
quaternions in Series 414.
(1). The vector σ of a centre s of curvature of the given surface,
answering to a given point P thereon, may (by (W1) and (X1)) be ex-
pressed by the equation,
which may be regarded also as a general form of the Vector Equation
of the Surface of Centres, or of the locus of the centre s: the vari-
able vector p of the point P of the given surface being supposed (p. 501)
to be expressed as a vector function of two independent and scalar
variables, whereof therefore v, r, and a become also functions,
although the two last involve an ambiguous sign, on account of the
Two Sheets of the surface of centres.
(j). The normal at s, to what may be called the First Sheet, has
the direction of the tangent to what may (on the same plan) be
called the First Line of Curvature at P; and the vector v of the point
corresponding to s, on the corresponding sheet of the Reciprocal (comp.
pp. 507, 508) of the Surface of Centres, has (by p. 684) the expres
do, du denoting any infinitesimal variations of the vectors σ and v,
consistent with the equations of the surface of centres and its recipro-
cal, or any linear and vector elements of those two surfaces, at two
corresponding points; we have also the relations (pp. 684, 685),
in which w is a variable vector, represents (p. 684) the normal plane
to the first line (j) of curvature at P; or the tangent plane at s to the
first sheet of the surface of centres: or finally, the tangent plane to
that developable normal surface (g), which rests upon the second line of
curvature, and touches the first sheet along a certain curve, whereof we
shall shortly meet with an example. And if v be regarded, comp. (i),
as a vector function of two scalar variables, the envelope of the variable
plane (E2) is a sheet of the surface of centres; or rather, on account of
the ambiguous sign (i), it is that surface of centres itself: while, in
like manner, the reciprocal surface (j) is the envelope of this other
(m). The equations (W1), (W1') give (comp. the Note to p. 684),
combining which with (C2), we see that the equations (H1) of p. xxv.
are satisfied, when the derived vectors p' and o' are changed to the cor-
responding differentials, dp and do. The known theorem (of Monge),
that each Line of Curvature is generally an involute, with the corre-
sponding Curve of Centres for one of its evolutes (400), is therefore in
this way reproduced: and the connected theorem (also of Monge),
that this evolute is a geodetic on its own sheet of the surface of centres,
follows easily from what precedes.
(n). In the foregoing paragraphs of this analysis, the given sur-
face has throughout been arbitrary, or general, as stated in (d) and
(9). But if we now consider specially the case of a central quadric,
several less general but interesting results arise, whereof many, but
perhaps not all, are known; and of which some may be mentioned
(o). Supposing, then, that not only dv=ødp, but also v=pp, and Spv=fp=1, the Index Surface (410, (d)) becomes simply (p. 670) the given surface, with its centre transported from o to P; whence many simplifications follow.
(p). For example, the semiaxes a1, as of the index curve are now equal (p. 681) to the semiaxes of the diametral section of the given surface, made by a plane parallel to the tangent plane; and Tv is, as in 409, the reciprocal P-1 of the perpendicular, from the centre on this latter plane; whence (by (X1) and Xi")) these known expressions for the two curvatures result:
(q). Hence, by (e), if a new surface be derived from a given central quadric (of any species), as the locus of the extremities of normals, erected at the centre, to the planes of diametral sections of the given surface, each such normal (when real) having the length of one of the semiazes of that section, the equation of this new surfacet admits (p. 683) of being written thus:
(r). Under the conditions (o), the expression (C2) for a gives (p. 684) the two converse forms,
and therefore (p. 689), by (d), (p), and by the theory (407) of confocal surfaces,
if 2 be formed from by changing the semiaxes abe to a2b2c2; it being understood that the given quadric (abe) is cut by the two confocals (abic) and (a2b2c2), in the first and second lines of curvature through the given point P: and that σ is here the vector of that first centre s of curvature, which answers to the first line (comp. (j). Of course, on the same plan, we have the analogous expression,
Throughout the present Series 412, we attend only (comp. (a)) to the curvatures of the two normal sections of a surface, which have the directions of the two lines of curvature: these being in fact what are always regarded as the two principal curvatures (or simply as the two curvatures) of the surface. But, in a shortly subsequent Series (414), the more general case will be considered, of the curvature of any section, normal or oblique.
+ When the given surface is an ellipsoid, the derived surface is the celebrated Wave Surface of Fresnel: which thus has (H2) for a symbolical form of its equation. When the given surface is an hyperboloid, and a semiaxis of a section is imaginary, the (scalar and now positive) square, of the (imaginary) normal erected, is still to be made equal to the square of that semiaxis.