19. The sum of the reciprocals of the squares of the perpendiculars from the centre upon three conjugate tangent planes is constant.

20. Cones are drawn, touching an ellipsoid, from any two points of a similar, similarly situated, and concentric ellipsoid. Shew that they intersect in two plane curves.

Find the locus of the vertices of the cones when these plane sections are at right angles to one another.

21. Find the locus of the points of contact of tangent planes which are equidistant from the centre of a surface of the second order.

22. From a fixed point A, on the surface of a given sphere, draw any chord AD; let D' be the second point of intersection of the sphere with the secant BD drawn from any point B; and take a radius vector AE, equal in length to BD', and in direction either coincident with, or opposite to, the chord AD: the locus of E is an ellipsoid, whose centre is 4, and which passes through B. (Hamilton, Elements, p. 227.)

23. Shew that the equation

72 (e2-1) (e + Saa') = (Sap)2 - 2e SapSa'p+ (Sa'p)2 + (1 —e2) p2, where e is a variable (scalar) parameter, and a, a' unit-vectors, represents a system of confocal surfaces. (Ibid. p. 644.)

24. Shew that the locus of the diameters of

Spop = 1

which are parallel to the chords bisected by the tangent planes to the cone Spyp = 0

25. Find the equation of a cone, whose vertex is one summit of a given tetrahedron, and which passes through the circle circumscribing the opposite side.

26. Shew that the locus of points on the surface

Spopp = 1,

the normals at which meet that drawn at the point p=,

27. Find the equation of the locus of a point the square of whose distance from a given line is proportional to its distance from a given plane.

28. Shew that the locus of the pole of the plane

is a sphere, if a be subject to the condition.

Sap 2a = C.

29. Shew that the equation of the surface generated by lines drawn through the origin parallel to the normals to

Spp-1p = 1

along its lines of intersection with

Sp (4+ k)1p = 1,

is

w2-kSw (4+ k)−1∞ = 0.

30. Common tangent planes are drawn to

2SλpSup+(p-Sλμ) p2 = 1, and Tph,

find the value of h that the lines of contact with the former surface

may be plane curves.

Discuss the case of

What are they, in this case, on the sphere? p2 - S2λμ = 0.

32. Tangent cones are drawn from every point of S(p-a) (p-a) = n2,

to the similar and similarly situated surface

Spop = 1,

shew that their planes of contact envelop the surface

(Sapp-1)2= n2Sppp.

33. Find the envelop of planes which touch the parabolas

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.

35. Discuss the surfaces whose equations are

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

37. If a1, a2, as be vector conjugate diameters of

CHAPTER IX.

GEOMETRY OF CURVES AND SURFACES.

279.] We have already seen (§ 31 (7)) that the equations

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 too 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, s, 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

where

p = $8,
ρ

Of

is a vector function of the length, s, of the curve. course it is only a linear function when the equation (as in § 31 (7)) 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 ex

tremity of p.

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

p's + p ́s òs + &c.

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 $'s+$"8d8+......, 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 os 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.

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

δα 68

p'p"" + p′′2 = c £ = cc1Up′′=c1p′′,

if c1 represent the tortuosity.

Integrating we get p'p" = c1p' + ß,

(1)

where ẞ is a constant vector. Squaring both sides of this equation, we get

(for by operating with S.p' upon (1) we get +c1 =

=

Spp'),

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

1

p = ¿cos.sTB+nsin.sTB-T2 (8B+Vaß), ......

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

n

which requires that

The farther test, that

-1=

c

(2)

(3)

Tß2(§2sin2.sTß+n2cos2. 8TB-2 S§nsin.sTßcos.sT3) — c2+c2

This requires, of course,

2

с

Sεn = 0,

Tε = Tŋ= c2 + c22

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 tangent to the curve p = $s

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

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

viously

-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 8 from any point of the curve. The equation of an involute which commences at this assumed point is

☎ = p-sp'.