Elements of Quaternions

62 = φιν = φιφρ,

for the vector of the second centre.

Pages.

(K2')

(s). These expressions for σ1, 02 include (p. 689) a theorem of Dr.
Salmon, namely that the centres of curvature of a given quadric at a
given point are the poles of the tangent plane, with respect to the two
confocals through that point; and either of them may be regarded,
by admission of an ambiguous sign (comp. (i)), as a new Vector Form*
of the Equation of the Surface of Centres, for the case (o) of a given
central quadric.

(t). In connexion with the same expressions for σ1, 2, it may be
observed that if ri, 2 be the corresponding values of the auxiliary
scalar in (c), and if 7, 7' still denote the unit tangents (g) to the
first and second lines of curvature, while abc, aibici, and a2b2c2 retain
their recent significations (r), then (comp. pp. 686, 687, see also p.
652),

and

r1 =fr=fUdp = (a2 — a22)-1 = &c.,
r2=fr' =ƒUvdp= (a2 — a12)-1 = &c.;

(L2)

(L2')

and σ2 with α1,

this association of r1 and σ with a2, &c., and of r2
&c., arising from the circumstance that the tangents and r' have re-
spectively the directions of the normals v2 and v1, to the two confocal
surfaces, (a2b2c2) and (a1bıcı).

(u). By the properties of such surfaces, the scalar here called r2 is
therefore constant, in the whole extent of a first line of curvature;
and the same constancy of r2, or the equation,

df Uvdp = 0,

may in various ways be proved by quaternions (p. 687).

(M2)

(v). Writing simply r and ✈ for r1 and r1⁄2, so that r' is constant,

but variable, for a first line of curvature, while conversely r is con-

stant and variable for a second line, it is found (pp. 684, 685, 586),

that the scalar equation of the surface of centres (i) may be regarded

as the result of the elimination of -1 between the two equations,

1=S.o (1 + r-1p)-2pσ, (N2), and 0=S.σ (1+r1p)-3p2σ; (N2′)

whereof the latter is the derivative of the former, with respect to the

scalar 1. It follows (comp. p. 688), that the First Sheet of the Sur-

face of Centres is touched by an Auxiliary Quadric (N2), along a Quartic

Curve (N2) (N2'), which curve is the Locus of the Centres of First Cur-

vature, for all the points of a Line of Second Curvature; the same

sheet being also touched (see again p. 688), along the same curve, by

the developable normal surface (2), which rests on the same second line :

with permission to interchange the words, first and second, through-

out the whole of this enunciation.

(w). The given surface being still a central quadric (0), the vec-

tors p, σ, v can be expressed as functions of v (comp. (j) (k) (1)),

* Dr. Salmon's result, that this surface of centres is of the twelfth degree, may
be easily deduced from this form.

CONTENTS.

and conversely the latter can be expressed as a function of any one of the former; we have, for example, the reciprocal equations (p. 685),

= (1 + r1p)-2 po;

(02)

o = (1 + r−16)2 4-1v, (02), and v= from which last the formula (N2) may be obtained anew, by observing (4) that Sov = 1. Hence also, by (r), we can infer the expres

xxxix Pages.

sions,

p = (p −1 + r1) v = 2 ̄1 v, (P2), and v =

-1

= P2 p = v2 ;

(P2')

and in fact it is easy to see otherwise (comp. p. 645), that v2 || 7 || v, and Spr2 = 1 Spv, whence v2 v as before.

(z). More fully, the two sheets of the reciprocal (j) of the surface of centres may have their separate vector equations written thus,

and the scalar equation† of this reciprocal surface itself, considered as including both sheets, may (by page 685) be thus written, the functions ƒ and F being related as in 408, (b),

with several equivalent forms; one way of obtaining this equation being the elimination of between the two following (same p. 685):

Fv+r1 v2 = 1, (Q2); fv+rv2 = 0.

(Q2')

(y). The two last equations may also be written thus, for the first sheet of the reciprocal surface,

F2 vi=1, (R2), and ƒʊv1 =r,

in which (comp. pp. 685, 689),

F2v=Svp21v= Sv (p−1 +r1) v ;

(R2)

(R2")

and accordingly (comp. pp. 483, 645), we have F2v2 Fv= 1, and ƒʊn=fr=r.

(2). For a line of second curvature on the given surface, the scalar r is constant, as before; and then the two equations (Q2), (Q2"), or (R2), (R2), represent jointly (comp. the slightly different enunciation in p. 688) a certain quartic curve, in which the quadric reciprocal (R2), of the second confocal (a2 b2 c2), intersects the first sheet (y) of the Reciprocal Surface (Q2); this quartic curve, being at the same time the intersection of the quadric surface (Q2′) or (R2), with the quadric cone (Q2′′) or (R2′), which is biconcyclic with the given quadric, fp=1.

* The equation v = v2, = the normal to the confocal (a2 b2 c2) at P, is not actually given in the text of Series 412; but it is easily deduced, as above, from the formulæ and methods of that Series.

+ The equation (Q2) is one of the fourth degree; and, when expanded by coordinates, it agrees perfectly with that which was first assigned by Dr. Booth (see a Note to p. 685), for the Tangential Equation of the Surface of Centres of a quadric, or for the Cartesian equation of the Reciprocal Surface.

ARTICLE 413.-On the Measure of Curvature of a Surface,

The object of this short Series 413 is the deduction by quaternions,

somewhat more briefly and perhaps more clearly than in the Lectures,

of the principal results of Gauss (comp. Note to p. 690), respecting

the Measure of Curvature of a Surface, and questions therewith con-

nected.

(a). Let P, P1, P2 be any three near points on a given but arbitrary

surface, and R, R1, R2 the three corresponding points (near to each other)

on the unit sphere, which are determined by the parallelism of the radii

OR, OR, OR2 to the normals PN, PIN1, P2 N2; then the areas of the two

small triangles thus formed will bear to each other the ultimate ratio

whence, with Gauss's definition of the measure of curvature, as the

ultimate ratio of corresponding areas on surface and sphere, we have, by

the formula (Z1) in 412, (ƒ), his fundamental theorem,

Measure of Curvature = R1 1 R2,

= Product of the two Principal Curvatures of Sections.

(S2')

(b). If the vector p of the surface be considered as a function of

two scalar variables, t and u, and if derivations with respect to these

be denoted by upper and lower accents, this general transformation

results (p. 691),

with a verification for the notation pqrst of Monge.

(c). The square of a linear element ds, of the given but arbitrary

surface, may be expressed (p. 691) as follows:

and with the recent use (b) of accents, the measure (T2) is proved
(same page) to be an explicit function of the ten scalars,

e,f,g; é,f,g'; fog.; and e,,-2f+g";

(U2')

the form of this function (p. 692) agreeing, in all its details, with the
corresponding expression assigned by Gauss.*

(d). Hence follow at once (p. 692) two of the most important
results of that great mathematician on this subject; namely, that
every Deformation of a Surface, consistent with the conception of it as
an infinitely thin and flexible but inextensible solid, leaves unaltered,

References are given, in Notes to pp. 690, &c. of the present Series 413,
to the pages of Gauss's beautiful Memoir, " Disquisitiones generales circa Superfi-
cies Curvas," as reprinted in the Additions to Liouville's Monge.

CONTENTS.

Is the Measure of Curvature at any Point, and IInd, the Total
Curvature of any Area: this last being the area of the corresponding
sertion (a) of the unit-sphere.

By a suitable choice of t and u, as certain geodetic co-ordinates,
the expression (2) may be reduced (p. 692) to the following,

where t is the length of a geodetic arc AP, from a fixed point a to a

variable point P of the surface, and u is the angle BAP which this

variable arc makes with a fixed geodetic AB: so that in the immediate

neighbourhood of A, we have n=t, and n' = Dm = 1.

(). The general expression (c) for the measure of curvature takes

thas the very simple form (p. 692),

xli
Pages.

Eis area being bounded by two geodetics, AP and AQ, which make with

each other an angle Au, and by an arc ro of an arbitrary curve on

the given surface, for which t, and therefore n', may be conceived to

be a given function of u.

(9). If this arc pa be itself a geodetic, and if we denote by v the

variable angle which it makes at P with AP prolonged, so that tan v
=nds: dt, it is found that don'du; and thus the equation (V2')
conducts (p. 693) to another very remarkable and general theorem of
Gauss, for an arbitrary surface, which may be thus expressed,

Total Curvature of a Geodetic Triangle ABC=A+B+C−π, (V1⁄2′′)

what may be called the Spheroidal Excess of that triangle, the total
area (47) of the unit-sphere being represented by eight right angles:
with extensions to Geodetic Polygons, and modifications for the case of
what may on the same plan be called the Spheroidal Defect, when the
taco curratures of the surface are oppositely directed.

ARTICLE 414.-On Curvatures of Sections (Normal and Oblique)
of Surfaces; and on Geodetic Curvatures,

(a). The curvatures considered in the two preceding Series hav-

ing been those of the principal normal sections of a surface, the present

Series 414 treats briefly the more general case, where the section is

made by an arbitrary plane, such as the osculating plane at p to an

arbitrary curve upon the surface.

(b). The vector of curvature (389) of any such curve or section

being (-)-1=D,3p, its normal and tangential components are found

to be (p. 694),

Αν

and

(W2')

the former component being the Vector of Normal Curvature of the

694-698

Surface, for the direction of the tangent to the curve: and the latter

being the Vector of Geodetic Curvature of the same Curve (or section).
(c). In the foregoing expressions, σ and are the vectors of the

points s and x, in which the axis of the osculating circle to the curve

intersects respectively the normal and the tangent plane to the sur-

face (p. 694); s is also the centre of the sphere, which osculates to

the surface in the direction dp of the tangent; 01, 02 are the vectors

of the two centres S1, S2, of curvature of the surface, considered in Se-

ries 412, which are at the same time the centres of the two osculating

spheres, of which the curvatures are (algebraically) the greatest and

least and v is the angle at which the curve here considered crosses

the first line of curvature.

(d). The equation (W2) contains a theorem of Euler, under the

form (p. 695),

RRcos2 v + R2-1 sin2 v;

(W2")

it contains also Meusnier's theorem (same page), under the form
(comp. 412, (h)) that the vector of normal curvature (b) of a surface,
for any given direction, is the projection on the normal v, of the vector
of oblique curvature, whatever the inclination of the plane of the sec-
tion to the tangent plane may be.

(e). The expression (W2'), for the vector of geodetic curvature, ad-

mits (p. 697) of various transformations, with corresponding expres-

sions for the radius T(p − ) of geodetic curvature, which is also the

radius of plane curvature of the developed curve, when the developable

circumscribed to the given surface along the given curve is unfolded

into a plane and when this radius is constant, so that the developed

curve is a circle, or part of one, it is proposed (p. 698) to call the given

curve a Didonia (as in the Lectures), from its possession of a certain iso-

perimetrical property, which was first considered by M. Delaunay,

and is represented in quaternions by the formula (p. 697),

8fS (Uv.dpop)+co (Tdp = 0;
'dp=V(Uv.dUdp),

(X2)