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therefore the direction-cosines of the principal normal are, by
reason of (31), cos sin <f>, and 0. The principal normal is
therefore perpendicular to the axis of z, and coincident with
the radius of the base-cylinder drawn to the point (x, y, z).

348.] In connexion with the subject of the osculating plane, it
is convenient to determine the analytical condition, that a curve
in space may be wholly in one plane; or in other words, that
every four consecutive points on the curve may be in one plane.

Let the equation to the plane be A# + By-fcz=n; then

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the geometrical meaning of which condition will be explained

349.] Of lines which can be drawn on a surface, and which
are therefore generally curves of double curvature, two classes
require notice in this place; although they will be discussed at
greater length in future parts of our Treatise; and when we
have more means at our command.

The first are geodesic lines, or geodesies as they are often called; they are those lines on a surface at all points of which the principal normal is coincident with the normal to the surface. And therefore their differential equations are

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Hereafter it will be seen that they are the shortest or the longest lines which can be drawn from one point on a surface to another. And as they are manifestly of great importance, from this point of view, in geodesy, so have they therefrom derived their name.

The second lines are lines of greatest slope, (lignes de plus grande pente of M. Monge); that is, if a surface is referred to three coordinate planes, one of which, say that of xy, is horizontal, the line of greatest slope starting from a given point on the surface is that curve each element of which makes with the plane of xy a greater angle than any other element on the surface abutting at the same point: and thus, since all the tangent lines at any point of a surface lie in the tangent plane at that point, that line which is perpendicular to the intersection of the tangent plane with the plane of xy makes the greatest angle with the plane of xy, and is therefore the line of greatest slope.

Let F (x, y, z) = 0 be the equation to the surface; then the equation to the tangent plane is

(f_,)(g)+(,_,)Q + (f_,)(g) = 0i m

the intersection of this with the plane of xy is the line

and if dx, dy, dz are the projections on the axes of an element common to both the surface and the line of greatest slope, then as the projection of this on the plane of xy is perpendicular to (45) we have . , .

and this differential equation combined with the equation to the surface will give the equations to the line of greatest slope. Let the surface be a sphere of radius a; then

r(x,y,z) = x* + y2 + z* a2 = 0;

so that (46) becomes xdy—ydx = 0;

x c t
y ~ b'

if the initial values of x and y are c and b; that is, the line of greatest slope is a meridianal arc.



350.]* In the present Chapter I propose to consider a few simple properties of surfaces, which are generated by straight lines and circles moving according to given laws; which lines, as they produce the surface, are called generators. The general theory is as follows:

Suppose that we have two equations involving x, y, z and two constants Cj and c2, and that they are of the forms,

Fi(#, J/,«) = Ci, r2(x,y,z) = cs; (1)

each of which represents a surface; and they, when taken conjointly, represent the line of intersection of the two surfaces. But if ci and r2 arc variable parameters, and dependent on each other by means of another equation,

/(ci, c,) = 0, (2)

then, as Ci and c% vary, the line of intersection of the two surfaces (1) varies, and by a continuous variation generates a surface, the equation of such a surface being found by the substitution of (1) in (2), whereby we have

/(*!, F2) = 0. (3)

The form however which such problems actually assume is generally somewhat different: a geometrical condition is given which is equivalent to the equation (2); thus, for instance, the generator may be a straight line which is to pass through a given curve, and move parallel to itself, or be parallel to a given plane and pass through two given lines, in which cases the curves through which the generator passes are called directors. The process of elimination is as follows:

* To those who desire further information on subjects connected with the discussions of the present Chapter I must recommend "Application d'Analyse a la Geometric par G. Monge; 5mc edition, par M. Liouville, Paris, 1850."

Let (1) be the equations to the generator involving two independent variable parameters, r, and c2; and let the equations to the director be,

*i (x, y, a) = 0, *i (x, y, z) = 0; (4)

then, as the generator is to pass through the director, x, y, z are at that common point the same in (1) and (4); eliminating therefore x, y, z, which refer to that common point, between these four equations, there will result a relation between Cj and c2 of the same form as (2), in which they must be replaced by their values in (1), and the resulting equation between x, y, z is that to the surface.

Again, the condition to which the generator is subject frequently is that it should circumscribe a given surface: the generator therefore must touch the given surface at their common point. Let the equation to the surface which is to be

circumscribed be n

w = 0; (5)

the direction-cosines of its normal at any point are proportional

(£>• 0- O

and the direction-cosines of the tangent of the generator are, Art. 341, equation (6), proportional to

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which we will symbolize respectively by P, Q, R; and as these lines are to be perpendicular to each other at the point of contact, so that the generator may touch the director-surface, we have the condition

at their common points, which is the equation to the curve of contact; and its intersection with (5) gives us a director through which the generator is to pass. Between therefore (1), (5) and (8), we may eliminate the coordinates which refer to their common points of contact, and get a relation between cj and c.. for which we may substitute the) general values of the coordinates given by (1). I propose now to consider those properties of the surfaces thus generated which the Differential Calculus enables us to elucidate.

Section 1.—On surfaces generated by the motion of straight


351.] Surfaces generated by the motion of straight lines are generally termed ruled surfaces (surfaces r(gKes), and of them there are two distinct classes: according as two consecutive generating lines do or do not intersect each other; or in other words, according as two consecutive generators are in the same or in different planes. Surfaces of the former class are termed developable, and those of the latter skew surfaces (surfaces gauches.).

The equations to a straight line being


six constants are apparently involved; of which however only four are indeterminate, because the equations can be put into the forms

* = «• + «,) (10) y = 0z + b; S

and of these four variable parameters, two fix the direction of the line and two fix its position. To eliminate these and to find the equation to the surface, five conditions are required, two of which are of necessity the equations (10) of the generator, and the other three are indeterminate, and may be given by means of the equations of three directors. Hence no ruled surface can in general have more than three directors; and to determine the surface from the equations to the generator, such conditions, or others equivalent to them in number, must be given. In general the directors may be such that two consecutive generators do not intersect, in which case the surface generated is skew; when however two successive generators intersect, the analytical condition of this being the case satisfies one of the relations which are required amongst the parameters, and leaves only two to be satisfied by the equations of

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