359.] On the equation of developable surfaces.

As any and every two consecutive generators of a developable surface intersect, and as these two, as shewn in Art. 352, lie in one plane, it is convenient to consider such a surface as formed by the continual intersection of planes drawn according to a given law. Now the general equation to a plane involves only three independent constants, and may be put in the form,

AX + BY + CZ = 1.

Suppose each of these constants to be a function of a variable parameter a, viz. let a = f(a), B = $(a), c = (a); then, as a continuously varies, the plane will have different positions, any two consecutive ones of which will intersect in a straight line; and will, as its position varies, generate a developable surface, of which the straight line of intersection of two consecutive planes will be the generator; thus the equation to one of the planes will be

xf(a) + y $(a) + z ¥ (a) = 1.

And if we differentiate it in reference to a, we have

x f'(a) + y p ́(a) + z √(a) = 0,

(48)

(49)

which is the equation to another plane; and the line of intersection of the two is the generator of the developable surface. From these two eliminating a, we shall obtain an equation in terms of x, y, z which will be that to the required surface: but this cannot be determined in the general case, that is, so long as the functions involved in (48) and (49) are undetermined. If however we eliminate the functions by differentiation, we shall get a differential equation which will give a property common to all developable surfaces.

360.] To find the differential equation to developable surfaces.

In (48) and (49), a is a function of x, y, z. Taking then the partial differentials of (48), we have

{xƒ'(a) + y$'(a) + zy' (a)} (da) +ƒ(a) + ↓ (a)

dz dx

= 0,

From these last equations, eliminating a, we have a relation of

and this is the general differential equation to developable surfaces.

Another differential equation equivalent to (51) may be found as follows:

Suppose the equation to the surface to be

F(x, y, z) = 0;

v dx + v dy + w dz = 0;

and u, v, w to be its partial derived-functions; then, as in Art. 332,

(52)

and suppose the equation to the plane by the consecutive intersection of which the surface is formed to be

AX + BY + CZ = 1;

(53)

and as this is a tangent plane, see Art. 352,

Differentiating again (52) and (54), since the tangent plane

touches the surface along the generating line,

v d2x + v d2y + w d2z + dv dx + dv dy + dw dz = 0, (56)

A d2x + B d2y + c d2% = 0;

whence replacing u, v, w from (55) in terms of A, B, C,

λ {▲ d2x + в d2y + c d2z} + dv dx + dv dy + dw dz = 0;

...

du dx + dv dy + dw dz = 0,

which is in fact identical with equation (49).

Comparing this with (52), we have

(57)

(58)

(59)

Let u, v, w, u', v', w' represent the several second partial

derived-functions of r (x, y, z), viz.

μv = dv = u dx+w'dy + v'dz,

μv = dv=w'dx + v dy + udz,

from which, and from (52), eliminating μ, dx, dy, dz, we have v2 (v w — u'2) + v2 (wu — v′2) + w2 (uv — w ́2)

+2vw (v'w' — uu') + 2wv (w'u' — vv')+2v v (u'v' — ww') = 0; (62) which result may easily be shewn to be identical with (51) by writing the equations in the form

F(x, y, z) = z − f(x, y) = 0.

and (51) takes the form (62), when (d),

are replaced by their values given in Art. 83.
are of course satisfied by the equations to the
the cone.

361.] Since every two consecutive generating lines of a developable surface intersect, a curve is formed, after the manner of an envelope, see Chapter XIII, Section 2, by the continual intersection of all these; and this must be a curve of double curvature, otherwise all the lines would be in one plane, and the developable surface would be only a plane. This curve bears the name of Edge of Regression (Arête de Rebroussement), and the generator of the surface is plainly always a tangent to it. Its equations may be found as follows:

Equations (48) and (49), if a is considered constant, are, taken together, the equations to a generating line of the developable surface, and therefore, from what has just been said, to the line whose envelope has to be determined: and the equations to which may therefore be found by making a to vary. Differentiating therefore (49) with respect to a, we have

x ƒ”(a)+yp′′(a) + z√”(a) = 0;

(63)

and eliminating (a) between this (48) and (49), we shall get two equations in terms of x, y, and z which are those to the edge of regression. This line, as is plain from its mode of

generation, bounds the developable surface towards one side of space; and on the other side the surface is continued.

362.] Hence also we arrive at a new conception of a developable surface; it is generated by a tangent of a curve of double curvature which moves continuously along the curve. Also since the osculating plane is that which contains two consecutive tangents, it may be conceived of as formed by the continuous intersection of such osculating planes. Suppose then that the equations to a curve of double curvature are given in the forms, see Ex. 2, Art. 347,

then, by equations (2), Art. 341, the equations to the generating line are

E-f(a) η-φ (α) 5— (a).

=

f'(a)

'(a)

=

'(a)

;

(66)

from which the equation to the surface will be found by the elimination of a.

Similarly also will developable surfaces be formed by the intersection of normal planes of a curve of double curvature; for suppose the equations to the curve to be of the form (64), then the equation to the normal plane is,

{§−ƒ(a)}ƒ'(a) + {n−¢ (a) } p′(a) + {5−↓ (a)} \'(a) = 0; (67) an equation of the form (48), and therefore manifestly that of a developable surface. Fig. 129 indicates the mode of generation of such surfaces and edges of regression.

363.] Ex. 1. To find the equation to the surface generated by tangents to the helix; or, which is the same thing, formed by the continuous intersection of osculating planes.

By Ex. 2, Art. 347, the equation to the osculating plane is

which is the equation to the developable helicoid, or screwsurface; the edge of regression of which is the helix itself.

Similarly will the equation to the surface formed by the intersection of consecutive normal planes to the helix be found to be

Ex. 2. If the equations to a curve of double curvature are

the student will without difficulty find the following equation to the surface formed by the intersection of normal planes;

which is manifestly the equation to a cone; the ambiguity of sign in the second term depending on k2 being greater or less than 62.

A further inquiry into the properties of developable surfaces is beyond the scope of the present treatise, but I cannot refrain from recommending the reader to study the works of Monge and Dupin on these subjects: works as they are of such intrinsic merit that I cannot venture to characterize them, for it may be that my praise would be below their due, and thus only tend towards disparagement.

364.] On skew surfaces.

Skew surfaces, see Art. 351, are those ruled surfaces, any two consecutive generating lines of which do not intersect; in the complete equations of such generators, (9) and (10) Art. 351, three conditions are left undetermined, and these may be, that the generator shall meet three directors. It is also manifest geometrically that such conditions fix the generator; for take a point chosen arbitrarily on any one to be the vertex of a cone, from which conceive two conical surfaces to be described with the other two generators as their directors; then these cones