The geometrical meaning of this condition is, that the normal to the surface always meets the axis of revolution whose equations are (95).

A plane which passes through the axis of revolution is called a Meridian plane; and its line of intersection with the surface of revolution is called a Meridian curve.

371.] Ex. 1. To find the equation to a surface described by a straight line revolving about the axis of 2, which it does not meet.

Let the equations to the revolving line in a given position of it be

x'

-α y' - B

=

=

then, if x and y are the current coordinates of the required surface, when + y22 = x2 + y2, '=z; but

x = a + = (x − y),

L

N

M

. . {a + = ( z − y) } 2 + { B + = (z−y)}2 = x2 + y2. (104)

N

N

which is the equation to a hyperboloid of revolution of one sheet, the centre of which is on the axis of z.

Ex. 2. To determine the conditions that the general equation of the second degree,

Ax2+By2+cz2 + 2A1 YZ +2 B1 zx+2c1 x Y

+2A2x + 2 Bq Y + 2 C2 z + K = 0, (105)

should express a surface of revolution.

The most general form that (98) admits of, so as to be an expression of the second degree is

(x-a)2+(y—b)2 + (z−c)2 = k2 (lx+my+nz)2;

(106)

expanding which and equating coefficients of the same powers of the variables with those of (105), we have

372.] On tubular surfaces.

Tubular surfaces are the envelopes of spheres of constant radii, whose centres are situated in a given curve, which is called the axis of the tube or canal.

The general theory of envelopes having been explained in Chapter XIII, it is unnecessary to enter on the subject at any great length, but one or two points require further elucidation.

Let F(x, yz, a) = 0 be the equation to the surface, involving x, y, z its current coordinates and a a variable parameter; and therefore representing a family of surfaces as a varies, and a particular individual of it for a particular value of a. Then the equation to the envelope is found by eliminating a between

dr

F = 0, and = 0,

da

(108)

whence will generally arise an equation in terms of x, y, z. Now although (108) thus give the equation to a surface, yet, if a is considered a constant in them, each when taken separately represents a surface, and when taken together they represent the line of intersection of two surfaces, and which is in general a curve of double curvature. To this line Monge has given the name of the characteristic. Thus if we conceive developable surfaces as formed by the intersection of consecutive planes, since two planes intersect in a straight line, a straight line is the characteristic, and is the generator of the developable surface.

Further, let us suppose, after the characteristic has been found, the variable a to vary again; hereby another characteristic will be formed, and will be determined by those two different equations, which will in general be different in form and position, and will cut the former one; thus an envelope will be formed of such characteristics, which will be an edge of regression, see Art. 361, and of course generally a curve of double curvature. Thus we shall have three equations,

from which eliminating a, we shall have two equations in terms of x, y, z, which will by their intersection give the edge of regression, and such as we have before met with in the case of developable surfaces. Fig. 129 will perhaps give a better notion of the formation of such a curve; but we shall return to the

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—p(a)}2 + {z—¥ (a) }2

The a-differential of this is

= a2.

(110)

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

— {(ƒ′(a))2 + (p′(a))2 + (y′(a))2} = 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

F1 (a, B, y) = 0, F2 (a, B, y) = 0;

(114)

and for the envelope of the spheres

(x − a) da + (y —ẞ) dB + (z−y) dy = 0,

(115)

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

544

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

(2 — a) (dz — da) + (y –B) (dy – d3) + (z−y)(z−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

X- α

U

y-B

=

=

V

a

=

W

(118)

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

U

Q dx = a dv

do

Q

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

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

therefore the group (119) becomes

dx {(au-q) (av — Q) (aw — Q)

-a2u'2 (au-Q) — a2 v'2 (av — q)—a2w'2 (aw − q) + 2 a3u'v'w' }

da

= a {u (av-Q) (aw—Q)—aw'v (aw—Q)

Q

— av'w (av−q) + a2 u' (u ́v + v'v + w ́w) −2 a2 u2v}; (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,

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

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

+ a2 {(u ́v + v'v+w'w)2 — 2 (u'2 v2 + v22 v2+w22w2)} = 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)3 + 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 7 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