(i.) If the equation of a central surface of the second order be put under the form fp= 1, where the function f is scalar, and homogeneous of the second dimension, then the differential of that function is of the form dfp 28.vdp, where the normal vector, v=pp, is a distributive function of p (homogeneous of the first dimension), dv = døp = ødp.

=

This normal vector v may be called the vector of proximity (namely, of the element of the surface to the centre); because its reciprocal, v1, represents in length and in direction the perpendicular let fall from the centre on the tangent plane to the surface.

(k.) If we make Sopp =ƒ(o, p), this function f is commutative with respect to the two vectors on which it depends, f(p, σ) = ƒ (o, p); it is also connected with the former function f, of a single vector p, by the relation, ƒ(p, p) = ƒp: so that fp = Sp¢p.

=

fdp Sdpdv; dfdp= 28.dvd'p; for a geodetic, with constant element,

this equation is immediately integrable, and gives const. Tv (fUdp)

=

product, PD.

=

reciprocal of Joachimstal's

(7.) If we give the name of "Didonia" to the curve (discussed by Delaunay) which, on a given surface and with a given perimeter, contains the greatest area, then for such a Didonian curve we have by quaternions the formula,

S.Uvdpop+cofTdp = 0,

where c is an arbitrary constant.

Derive hence the differential equation of the second order, equivalent (through the constant c) to one of the third order,

edp1.Urd Udp.

Geodetics are, therefore, that limiting case of Didonias for which the constant c is infinite.

On a plane, the Didonia is a circle, of which the equation, obtained by integration from the general form, is

p = ☎ +cUvdp,

being vector of centre, and c being radius of circle.

(m.) Operating by S.Udp, the general differential equation of the Didonia takes easily the following forms :

(n.) The vector w, of the centre of the osculating circle to a curve in space, of which the element Tap is constant, has for expression,

(0.) Hence, the radius of curvature of any one Didonia varies, in general, proportionally to the cosine of the inclination of the osculating plane of the curve to the tangent plane of the surface. And hence, by Meusnier's theorem, the difference of the squares of the curvatures of curve and surface is constant; the curvature of the surface meaning

here the reciprocal of the radius of the sphere which osculates in the reduction of the element of the Didonia.

(p.) In general, for any curve on any surface, if § denote the vector of the intersection of the axis of the element (or the axis of the circle osculating to the curve) with the tangent plane to the surface, then

Hence, for the general Didonia, with the same signification of the symbols,

} = p-cVvdp;

and the constant c expresses the length of the interval p-, intercepted on the tangent plane, between the point of the curve and the axis of the osculating circle.

(q.) If, then, a sphere be described, which shall have its centre on the tangent plane, and shall contain the osculating circle, the radius of this sphere shall always be equal to c.

(r.) The recent expression for έ, combined with the first form of the general differential equation of the Didonia, gives

(8.) Hence, or from the geometrical signification of the constant c, the known property may be proved, that if a developable surface be circumscribed about the arbitrary surface, so as to touch it along a Didonia, and if this developable be then unfolded into a plane, the curve will at the same time be flattened (generally) into a circular arc, with radius

= c.

24. Find the condition that the equation.

Sp (+)-'p = 1

may give three real values of ƒ for any given value of p. If f be a function of a scalar parameter έ, show how to find the form of this function in order that we may have

Prove that the following is the relation between ƒ and έ,

25. Show, after Hamilton, that the proof of Dupin's theorem, that "each member of one of three series of orthogonal surfaces cuts each of the other series along its lines of curvature," may be expressed in quaternion notation as follows:

=

If SvVv = 0, SvVv0, Sv"V" 0, and V.vvv 0,

=

represents the line of intersection of a cylinder and cone, of the second order, which have ẞ as a common generating line.

CHAPTER X.

KINEMATICS.

do

expressed by

dt

That is, if

hen a point's vector, p, is a function of the time t, we have seen (§ 35) that its vector-velocity is

or, in Newton's notation, by p.

p = $t

be the equation of an orbit, containing (as the reader may see) not merely the form of the orbit, but the law of its description also, then

p = 't

gives at once the form of the Hodograph and the law of its description.

This shows immediately that the vector-acceleration of a point's motion

or p,

dt2

Thus the fundamental

is the vector-velocity in the hodograph.

properties of the hodograph are proved almost intuitively.

336. Changing the independent variable, we have

dp ds

p =

ds dt

= vp,

if we employ the dash, as before, to denote

d

ds

This merely shows, in another form, that p expresses the velocity in magnitude and direction. But a second differentiation gives

ï = ip' + v2p".

This shows that the vector-acceleration can be resolved into two components, the first, vp', being in the direction of motion and