Elements of Quaternions

itself as a certain locus of chords (b), admits of many quaternion
transformations; for instance (see p. 675), it may be written thus,

ΒαρΔρ. Βα'ρΔρ

ΒαΔρ Βα ́Δρ

= 0,

(Vi)

pbeing the vector of the vertex P, and p + Ap that of any other point

r' 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

xxxiii

Pages.

* It might be natural to suppose, from the known general theory (410, (c))
of the two rectangular directions, that each such generatrix Pr' is crossed perpendi-
cularly, at every one of its non-umbilicar points P', 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 O

(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 r' of an umbilicar generatrix, it may be added that the cone

of chords (C), in 411, (6), is found to touch the quadric along that generatrix,

when its vertex is at any such point r'.

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-

teral hyperbola.

(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):-If
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,

(a). If o be the vector of the centre s of curvature of a normal

section of an arbitrary surface, which touches one of the two lines of

curvature thereon, at any given point r, we have the two fundamental

equations (p. 679),

the equation (W1") 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 r (|| dp) is a tangent to a line of curvature, while dv = φαρ,
as in (U1), the two values of r, which answer to the two rectangular
directions (T1") in 410, (c), are given (p. 680) by the expression,

in which g, λ, μ are, for any given point r, the constants in the equa-
tion (Ur") 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 || A or || μ,
and where consequently the index curve is a circle.
(d). At any other point of the given surface, which is as yet en-

tirely arbitrary, the values ofr may be thus expressed (p. 681),

11=81-2, 12,

(X1")

a1, a2 being the scalar semiaxes (real or imaginary) of the index curve

(defined, comp. 410, (d), by the equations Sp'φρ' = 1, δνρ' = 0).

Pages.

679-689

under the symbolical form,

Sv-1 (+r) y = 0;

CONTENTS.

(e). The quadratic equation, of which 1 and 12, or the inverse
squares of the two last semiaxes, are the roots, may be written (p. 683)

(Y1)

XXXV

Pages.

which may be developed (same page) into this other form,

r2 + rv -1 xv + Sv -1 ψν = 0,

(Y1)

the linear and vector functions, and x, being derived from the func-

tion o, on the plan of the Section III. ii. 6 (pp. 440, 443).

(f). Hence, generally, the product of the two curvatures of a sur-

which will be found useful in the following series (418), in connexion
with the theory of the Measure of Curvature.

(g). The given surface being still quite general, if we write

(p. 686),

τ = Udρ, τ' = U (vdp), (A2), and therefore rr' = Uv, (Α ́2)

so that and r' are unit tangents to the lines of curvature, it is easily

proved that

dr' = r Sr'dr, (B2), or that Vrdr'=0;

(B2)

this general parallelism of dr' to 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 be tangential to the

line.

(A). 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; 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 of a centres of curvature of the given surface,

answering to a given point r thereon, may (by (W1) and (X1)) be ex-

pressed by the equation,

σ=ρ+r-1ν;

(C2)

which may be regarded also as a general form of the Vector Equation
of the Surface of Centres, or of the locus of the centres: the vari-
able vector p of the point 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 o 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 atr; 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-
sion,

which may also be considered (comp. (i)) to be a form of the Vector
Equation of that Reciprocal Surface.

(k). The vector v satisfies generally (by same page) the equations
of reciprocity,

Συσ= $συ = 1, Sυδσ = 0, $σδυ = 0,

(D2')

δα, δυ denoting any infinitesimal variations of the vectors o 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),

$ρυ = 1, vv = 0, Σνυφυ = 0.

(1). The equation Sv (ω- ρ) = 0, or more simply,

Συω = 1,

(D2")

(E2)

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 u 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

plano

$σω = 1.

(E2')

(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 σ' are changed to the cor-

responding differentials, do 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

(g). 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

here.

Pages.

CONTENTS.

(0). Supposing, then, that not only dv=ødp, but also v = φρ, and Spv = fp = 1, the Index Surface (410, (d)) becomes simply (p. 670) the given surface, with its centre transported from o top; whence many simplifications follow.

(p). For example, the semiaxes a1, a2 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 X1")) 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 semiaxes 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 gives (p. 684) the two converse forms,

xxxvii Pages.

and therefore (p. 689), by (d), (p), and by the theory (407) of confocal surfaces,

(K2)

if $2 be formed from by changing the semiaxes abe to a26202; it being understood that the given quadric (abe) is cut by the two confocals (abıcı) and (a2b202), in the first and second lines of curvature through the given point: and that σ1 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.

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