32. Tangent cones are drawn from every point of

S(p-a)(p-a) = n',

to the similar and similarly situated surface

Spop = 1,

show that their planes of contact envelop the surface

(Sapp-1)= n2 Sppp.
(αφρ-

33. Find the envelop of planes which touch the parabolas

p = at2 +ßt, p = ar2+YT,

where a, ẞ, y form a rectangular system, and t and ↑ are scalars.

34. Find the equation of the surface on which lie the lines of contact of tangent cones drawn from a fixed point to a series of similar, similarly situated, and concentric ellipsoids.

36. Show that the locus of the vertices of the right cones which touch an ellipsoid is a hyperbola.

CHAPTER IX.

GEOMETRY OF CURVES AND SURFACES.

279. We have already seen (§ 31 (7)) that the equations p = pt = Σ.af(t),

where a represents one of a set of given vectors, and ƒ a scalar function of scalars t and u, represent respectively a curve and a surface. We commence the present brief Chapter with a few of the immediate deductions from these forms of expression. We shall then give a number of examples, with little attempt at systematic development or even arrangement.

280. What may be denoted by t and u in these equations is, of course, quite immaterial: but in the case of curves, considered geometrically, t is most conveniently taken as the length, 8, of the curve, measured from some fixed point. In the Kinematical investigations of the next Chapter t may, with great convenience, be employed to denote time.

281. Thus we may write the equation of any curve in space as

p = $8,

where is a vector function of the length, s, of the curve. Of course it is only a linear function when the equation (as in § 31 (h)) represents a straight line.

282. We have also seen (§§ 38, 39) that

is a vector of unit length in the direction of the tangent at the

extremity of p.

At the proximate point, denoted by s+ds, this unit tangent vector becomes

But, because

we have

p's+$"s ds + &c.

To's 1,

=

S.p's "s = 0.

Hence "s is a vector in the osculating plane of the curve, and perpendicular to the tangent.

Also, if de be the angle between the successive tangents p's and 's+'s 88+

we have

so that the tensor of "'s is the reciprocal of the radius of absolute curvature at the point s.

283. Thus, if OP = 4s be the vector of any point P of the curve, and if C be the centre of curvature at P, we have

is the equation of the locus of the centre of curvature.

is the vector perpendicular to the osculating plane; and

is the tortuosity of the given curve, or the rate of rotation of its osculating plane per unit of length.

284. As an example of the use of these expressions let us find the curve whose curvature and tortuosity are both constant.

We have

curvature= To's = To" c.

=

where a is a unit vector perpendicular to the osculating plane.

where a is a constant quaternion. Eliminating p', we have

(2)

έ and ʼn being any two constant vectors. We have also by (2),

η

c

-1=TB'('sin'.8TB+n'cos2.8 TB-2 Sensin.8 TB cos.8 TB)

This requires, of course,

Sen = 0,

с

Tε = Tη = c2 + ci

so that (3) becomes the general equation of a helix traced on a right cylinder. (Compare § 31 (m).)

285. The vector perpendicular from the origin on the tan

To find a common property of curves whose tangents are all equidistant from the origin.

This equation shows that, as is otherwise evident, every curve on a sphere whose centre is the origin satisfies the condition. For obviously

-p2 = c2 gives Spp'= 0,

and these satisfy (1).

If Spp' does not vanish, the integral of (1) is

an arbitrary constant not being necessary, as we may measure s from any point of the curve. The equation of an involute which commences at this assumed point is

This includes all curves whose involutes lie on a sphere about the

origin.