Or taking the implicit form, viz. equation (83), and writing F for the derived-function of F,

the geometrical meaning of which is, that the normal to such a surface is perpendicular to the line drawn from the point of intersection perpendicular to the axis of z.

which surface is called the Skew Helicoid, and is that of the under surface of spiral staircases.

Ex. 2. Let the director be a circle whose plane is parallel to that of xz at a distance c from it, and whose centre is in the axis of y; see fig. 126.

Let radius of circle = a, and let the generator pass through and be perpendicular to the axis of x; OM = x, MN = y, NP; OM = x', ML = y', LQ = z.

which surface is known by the name of the Cono-Cuneus of Wallis, of which the figure contains but one-fourth, the remainder being in three other octants.

316.] To find the Equation to a surface generated by a straight line moving on two Directors and always parallel to a given plane.

It is manifest from the mode of generation, that a section of the surface made by a plane parallel to the given plane is a straight line.

Let Ax + By + cz = a be the equation to the plane cutting the surface and parallel to the given plane, and therefore having A, B, C constant, and a a variable parameter; and let the equation to another plane passing through the generating straight line of the surface be

A1x + B1Y + C12 = 0,

(90)

that is, conceive it to pass through the origin; then A1, B1, and C1 are variable and may be considered to be functions of a, so that A1 = f(a), B1 = (a), C1 = √(a);

and therefore the general equation to the surface is

xf(Ax+BY+Cz)+Y $ (AX +BY+CZ)+Z¥ (AX + By + cz) = 0. (91)

To find its differential equation: to simplify the process, suppose that the director plane is parallel to that of xy; then A = B = 0, and c = 1, and the equation becomes

xf(z) + y$(z) + z√(z) = 0,

which may be put into the form

zỰ(z)

(92)

SECTION 2.-On Surfaces generated by the Motion of Circles. 317.] On Surfaces of Revolution.

DEF. A surface of revolution is generated by a curve which revolves about a straight line called the axis, and every point of which describes a circle about the axis.

Hence, if such a surface is cut by a plane perpendicular to the axis of revolution, the section is the circumference of a circle whose centre is on the axis, and all points of which are consequently at equal distances from the axis.

Let the equations to the axis be

(a, b, c) being a given point through which it passes, viz. the point a in fig. 127, and l, m, n being its direction-cosines.

Let x, y, z be the current coordinates of the surface; then the equation to the plane passing through (x, y, z), and perpendicular to (95), is l x + my + nz = p.

(96)

Let BAQ be the axis of revolution, Oв the perpendicular from the origin on it, RP the generating curve; and suppose the equation of it to be given in the form

then (x − a)2 + (y—b)2 + (≈−c)2 = f(lx + my + nz), (98) and which is the general equation to surfaces of revolution. If the axis of revolution be that of z, then a = b = c = 0, l = m = 0, n = 1, and

x2 + y2 + z2 = f(z);

or, what is equivalent,

2= f(x2 + y2).

(99)

318.] To find the Differential Equation to Surfaces of Revolution.

Eliminating ƒ from (98) according to the method of Art. 89, we have

l − —

dz

m (x − a) — 1 (y — b) + {n (x − a) — 1 (z — c)} (day)

+ {m (z − c) − n (y − b) } (dz) = 0. (100)

dx

Or putting (98) in the form

F(x, y, z) = F{(x —− a)2 + (y—b)2 + (z−c)2, lx+my+nz} = 0, (101) and representing by r' the derived-function of F,

The geometrical meaning of which condition is, that the normal to the surface always meets the axis of revolution.

319.] Ex. 1. To find the Equation to a Surface described by a straight line revolving about the axis of z, which however it does not meet.

Let the equations to the revolving line be

•'. {a + — (≈−y)}3 + {ß + = (z−y)}2 = x2 + y2. (104)

Ex. 2. To determine the conditions that

Ax2 + By2+ Cz2 + 2A1yz + 2B1zx+2c1xy

+2A2x + 2B2Y + 2C2≈ + K = 0 (105)

should express a surface of revolution.

The most general form that (91) admits of, so as to be an expression of the second degree is

(x-a)2+(y-b)2 + (≈−c)2 = k2 (lx+my+nz); (106)

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

DEF.-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, y, z, 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

whence will generally arise an equation in terms of x, y, z.

(108)

Now although (108) thus gives the equation to a surface, yet, if a be 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; to this line Monge has given the name of characteristic. Thus conceiving of 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 of course the generator of the developable surface.

Further imagine that, after the characteristic has been found, the variable a varies again; we shall hereby have another