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-cUvdp;

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

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

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

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

If

Svdp = 0,

Sv'dp = 0, S.vv'dp=0

be integrable, and if

=

Svv 0, then Vudp 0, makes S.vv'dv = 0.

=

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

27. Two spheres are described, with centres at A, B, where OA = a, OB = ẞ, and radii a, b. Any line, OPQ, drawn from the origin, cuts them in P, Q respectively. Shew that the equation of the locus of intersection of AP, BQ has the form

V(a+aU (p-a)) (B+bU (p—ß)) = 0.

and therefore that the left side is a scalar multiple of V.aß, so that the locus is a plane curve.

Also shew that in the particular case

the locus is the surface formed by the revolution of a Cartesian oval about its axis.

CHAPTER X.

KINEMATICS.

dp

336.] WHEN a point's vector, p, is a function of the time t, we have seen (§36) that its vector-velocity is expressed by Newton's notation, by p.

or,

in

dt

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 = o't

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

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

d2p
dt2

or P,

is the vector-velocity in the hodograph. Thus the fundamental properties of the hodograph are proved almost intuitively.

337.] Changing the independent variable, we have

dp ds
ds dt

=

υρ',

if we employ the dash, as before, to denote

d ds

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

ï = vp + v2p".

This shews that the vector-acceleration can be resolved into two components, the first, vp', being in the direction of motion and dv equal in magnitude to the acceleration of the velocity, v or ; dt the second, v2p", being in the direction of the radius of absolute

curvature, and having for its amount the square of the velocity multiplied by the curvature.

[It is scarcely conceivable that this important fundamental proposition, of which no simple analytical proof seems to have been obtained by Cartesian methods, can be proved more elegantly than by the process just given.]

338.] If the motion be in a plane curve, we may write the equation as follows, so as to introduce the usual polar coordinates, r and 0,

20

Pra" B,

where a is a unit-vector perpendicular to, ẞ a unit-vector in, the plane of the curve.

Here, of course, r and 0 may be considered as connected by one scalar equation; or better, each may be looked on as a function of t. By differentiation we get

20

20

which gives, by inspection, the components of acceleration along, and perpendicular to, the radius vector.

339.] For uniform acceleration in a constant direction, we have at

where is the vector-velocity at epoch. This shews that the hodograph is a straight line described uniformly.

no constant being added if the origin be assumed to be the position of the moving point at epoch.

Since the resolved parts of p, parallel to ẞ and a, vary respectively as the first and second powers of t, the curve is evidently a parabola (§ 31 (ƒ)).

But we may easily deduce from the equation the following result,

the equation of a paraboloid of revolution, whose axis is a. Also

S.aßp = 0,

and therefore the distance of any point in the path from the point — ßa ̄1ß is equal to its distance from the line whose equation is

Thus we recognise the focus and directrix property.

340.] That the moving point may reach a point y we must have, for some real value of t,

Now suppose Tß, the velocity of projection, to be given v, and, for shortness, write for Uẞ.

is positive. Now, as TaTy is never less than Say, it is evident that 2-Say must always be positive if the roots are possible. Hence, when they are possible, both values of t2 are positive. Thus we have four values of t which satisfy the conditions, and it is easy to see that since, disregarding the signs, they are equal two and two, each pair refer to the same path, but described in opposite directions between the origin and the extremity of y. There are therefore, if any, in general two parabolas which satisfy the conditions. The directions of projection are (of course) given by the corresponding values of @.

341.] The envelop of all the trajectories possible with a given velocity, evidently corresponds to

(v2-Say)2-Ta2Ty2 = 0,

for then y is the vector of intersection of two indefinitely close paths in the same vertical plane.

is evidently the equation of a paraboloid of revolution of which the origin is the focus, the axis parallel to a, and the directrix plane at a distance

v2

Ta

All the ordinary problems connected with parabolic motion are easily solved by means of the above formulae. Some, however, are even more easily treated by assuming a horizontal unit-vector in