subject in Chapter XVIII, and discuss it in a particular case at a length which will remove many difficulties.

The complete investigation of tubular surfaces requires three equations to be found: (1) that of the envelope of all the spheres (2) that of the characteristic: (3) that of the edge of regression.

Let a the constant radius of the sphere. And let the equations to the axis be expressed in terms of a single variable parameter a, so that the equation to a sphere may be

{x −ƒ(a)}2 + {y—$(a)}2 + {z—¥(a)}2 = a2. (110)

The a-differential of this is

{x−f(a)} f'(a)+{y−p(a)} p′(a) + {z—¥ (a)} \'(a) = 0; (111) which, taken in combination with (110) when a is constant, represents the characteristic; and as (111) represents a plane, the characteristic is manifestly a great circle of the sphere.

Differentiating (111) again, we have

{x−ƒ(a)} ƒ" (a) + {y—p(a)} p′′(a) + {z−¥ (a)} 4′′(a)

— {(f'(a))2 + (p'(a))2 + (y' ́(a))"} = 0. (112)

By means of which and (110) and (111) if a is eliminated, there will be two equations in terms of x, y, z, which, taken in combination, are those to the edge of regression formed by the characteristics.

373.] Now all tubular surfaces have a common property, which may be expressed as a differential equation. This we proceed to find.

Let the equation to the surface be F(x, y, z) = 0, of which u, v, w are the partial derived-functions; and let the equation to the generating sphere be

(x-a)2+(y-3)2 + (≈—y)2 = a2.

(113)

Now (a, ß, y) being the centre of the sphere, and the centre being on a given curve, these quantities are connected by two equations of the form

da, dß, dy having two other relations given by (114), into

which however it is of no use generally to inquire further. Now differentiating (113) we have

( −a) (dữ —da) + (y —B) (dy — dB) + (≈ −y) (đã —dy) = 0 ; (116) and therefore by (115),

(x − a) dx + (y-ẞ) dy + (z―y) dz = 0;

(117)

which shews that a tangent to the sphere at the points of it which are common to the sphere and the envelope is perpendicular to the line drawn from that point to the centre of the sphere. This latter line therefore is the normal to the tubular surface; and as the direction-cosines of the normal of a surface are proportional to u, v, and w, we have

Now as the envelope touches each of the enveloped spheres, and as the envelope and enveloped sphere have one and the same tangent plane, so for a point common to the two surfaces x, y, z, u, v, w, are the same, whether we consider the point as belonging to the sphere or to the envelope. And therefore we may differentiate the equations (118) under this condition; whence we have

and using the notation of Art. 361, equations (61),

dv = u dx+w'dy + v'dz
dvw'dx+v dy + udz

dw = v'dx + u'dy + w dz

therefore the group (119) becomes

(au-q) dx + aw'dy +

av'dz

=

[da]

aw'dx + (av-Q)dy + au'dz

av'dx + au'dy + (aw-q)dz = a

Whence, by cross-multiplication,

da

dx {(au-q) (av — Q) (aw — Q)

— a3 u' 2 (au — Q) — a2 v′ 2 (a v — Q) — a2 w'2 (aw − q) + 2 a3u'v'w' }

da {v (av — q) (aw — q) — aw'v (aw — q)

-av'w (av-Q) + a2 u' (u ́v + v ́v+w'w) -2 a2u'2v}; (121) and similarly may the values of dy and dz be found. Multiplying through therefore by u, v, w, adding, and by means of (17), Art. 332,

v3 (av — Q) (aw — Q) + v2 (aw — Q) (au − q) + w2 (au — Q) (av — Q)

−2 a {u'vw (au — Q) + v' w u (av — Q) + w' v v (aw − q)}

+a2 {(u'v+v'v+w'w)2 — 2 (u'2 v2 + v22 v2 +w'2w2)} = 0. (122)

If a∞, this condition becomes identical with that given in equation (62) for developable surfaces.

If the director curve of the centre of the sphere is a plane curve in the plane of xy; then the equation to the sphere is

(x — §)2 + (y — n)2 + z2 = a2.

(123)

As the tubular surface is an envelope we may take partial differentials of this equation on the supposition that the terms involving the differentials of έ and ʼn vanish; so that

η

which is the differential equation of such tubular surfaces: and if it is expressed in terms of u, v, w, Q, it becomes

The geometrical meaning of (124) and (125) is, the part of the normal between the surface and the plane of xy is equal to the radius of the generating sphere.

PRICE, VOL. I.

4 A

374.] Examples of tubular surfaces.

Ex. 1. Let the axis of the tube be a straight line whose

whence έ, n, may be found in terms of x, y, z; and thence, substituting in the equation to the sphere,

{L(MY+NZ) −x (M2 + N2)}2 + {M (N≈ + Lx) −y (N2 +L2)}2

+{N(LX + My) —≈ (L2 + M2)}2 = a2 (L2 + M2 + N2)2.

Ex. 2. To find the equation to the surface of a circular ring. Let the director curve of the centre of the generating sphere be §2 + n2 = c2;

so that the equation of the sphere is

.. {{c+(a2 —≈2)} = cx,

n {c + (a2 — ~2) + }

•'. {c+(a2 — z2) §}2 = x2 + y2 ;

which is the equation to the required surface.

= cy;

CHAPTER XVII.

ON CURVATURE OF CURVES IN SPACE, AND ON

CERTAIN KINDRED AFFECTIONS.

375.]* CERTAIN principles, names, and modes of estimation which were discussed in Chapter XII, as to the curvature of curves, are stated with breadth sufficient to include kindred properties of curves in space; a difference however of great importance exists between the two classes, and which it is necessary at once to bring out into greater prominence. In the former case the whole of the curve lies in one plane, and the curve is therefore called a plane curve; in the present, although every two consecutive elements, or every three consecutive points, must be in one plane, viz. the osculating plane, yet the third element, or the fourth point, may be, and generally will be, in a different plane. For this reason such curves are called non-plane curves, and from this general property arise other affections of a more complex character, and which we proceed to inquire into.

Consider a portion of a curve in space, at no point of the part of which under investigation is there a point of abrupt termination, or of discontinuity, and at which the derived-functions of the equations to the curve are not indeterminate. Now as every three consecutive points must be in one plane, and as the mode of estimating curvature as explained in Art. 290 requires only three points in the curve's plane, the principles therein investigated are immediately applicable, and we propose to apply them by a similar process, viz. by drawing in the osculating plane, which is the plane containing three consecutive points, two consecutive normals, which will generally meet at a finite distance from the curve; the ratio of the infinitesimal

* For a most masterly exposition of the properties considered in this Chapter, and for geometrical proofs of them by the infinitesimal method, the reader is requested to consult a Memoir by M. de Saint Venant in "Trentième Cahier du Journal de l'École Royal Polytechnique."-Bachelier, Paris, 1845.