ay2+ bz2 = 0,

(44)

therefore equation (43) becomes

which is satisfied only by y = 0, ≈ 0; therefore (44) represents the axis of x, or the surface at the origin degenerates into a cuspal point formed round the axis of x.

Ex. 2. A surface is formed by the revolution of a parabola about an ordinate through its focus; it is required to find the nature of the points where it meets the axis of z.

The equation to the surface is

16m2 (x2+ y2)-(22-4m2)2 = 0;

whence it appears that x = y = 0, when z=2m; and at

again, and substituting in equation (42), we shall find

§2+n2−(5 ± 2m)2 = 0;

the equation to two right-angled circular cones, whose axis is the axis of z and vertices at distances + 2m from the origin.

If all the second differential-coefficients vanish at the point where the tangent plane is to be drawn, we must proceed to a third differentiation, or to the fourth term of the expansion in Art. 142; and thus we shall arrive at a cone of the third order.

CHAPTER XV.

APPLICATION OF THE DIFFERENTIAL CALCULUS TO PROPERTIES

OF CURVES IN SPACE.

340.] THE curves whose properties have been inquired into lie wholly in one plane; that is, all their elements and all their consecutive points have been entirely in the plane of xy; and we have considered them in reference to two fixed lines in that plane. It is manifest however that all curves are not subject to the restriction of having their elements in the same plane; there may be non-plane as well as plane curves, and as such they exist in space, and are conveniently referred to three coordinate axes meeting each other at right-angles and in one point; such are also called curves of double curvature, and for a reason which will be hereafter assigned. They may be determined in two ways: either by the intersection of two surfaces whose equations involving x, y, z are given, and therefore by the combination of these two equations; or, what amounts to the same thing, one of the variables, as e. g. z, may have been eliminated between these two equations, and an equation obtained involving only x and y, which will be the equation to the projection of the curve on the plane of xy; and so with the other variables; whereby three equations may be formed, each containing two variables, which will severally represent the projections of the curve on the coordinate planes, and any two of which equations will be sufficient to define the curve; and according as one or the other method is adopted the formulæ will assume different, though equivalent, shapes.

341.] To find the equations to a tangent line to a curve in space.

A tangent line is the straight line passing through two points on the curve which are infinitesimally near to each other.

Let έ, n, be the current coordinates to the tangent line, and first let the two points through which the line is to pass be at a finite distance as apart; and let them be (x, y, z), (x +AX, Y + ▲Y, +Az); then the equations to the line are

where

is the distance between the two points (x, y, z) and

(8, 7, (). When these two points become infinitesimally near to one another, the line becomes a tangent, and its equations become

where

E-x n-y
=
dx dy dz

ds =

=

(dx2 + dy2 + dz2),

(2)

and is the differential of the arc, or the length-element of the curve.

On comparing these equations with those of (4) in Art. 331, if λ, μ, v are the direction-angles of the tangent,

If the equations to the curve are two equations, say of the forms

dx

and

dz

dy
dz

f(x, z) = 0,

(y, z) = 0,

can be found by differentiation, and equations (2)

and (3) can be determined for the particular curve.

If the curve is determined by means of the equations to two surfaces of the forms,

F1 (x, y, z) = 0, F2 (x, y, z) = 0;

(4)

we have by elimination the following system of equations,

whence, multiplying the several terms of equality (2) by the several terms of this equality, dx, dy, dz will divide out, and

we shall have the equations to the tangent in terms of the partial differential-coefficients of the intersecting surfaces. Similarly may the direction cosines in (3) be determined.

342.] To find the equation to the normal plane to a curve in space.

The plane perpendicular to the tangent line, and passing through the point of contact, is called the normal plane. Let , 7, be its current coordinates, and (x, y, z) be the point of contact through which it passes; then, since it is to be perpendx dy dz

dicular to the line whose direction cosines are its equation is

(§—x) dx + (n−y) dy + (8—2) dz = 0.

ds' ds' ds'

(7)

343.] To find the equation to the osculating plane to a curve in space.

In curves such as we have discussed in previous Chapters, all the points lie in one plane; and therefore the curves are called plane curves. This property however does not hold good for all curves in space; although every three consecutive points must be in one plane, yet the fourth may be out of it; or in other words, every two consecutive tangents are in the same plane, but the next consecutive tangent is in general in a different one; our object is to determine the equation to the plane which contains two consecutive tangents, and which is called the osculating plane, and is defined as follows:

The osculating plane is the plane containing three consecutive points on a curve.

Let the equation to the plane be

and let it pass through the three points on the curve (x, y, z), (x+dx, y+dy, z+dz), (x+2dx + d2x, y +2dy + d2y, z+2 dz+d2z); whence we have

Ax+BY+CZ = D,

(9)

A

=

B

dy d2z-dz d2y dz d2x-dx d2z

C

dx d3y - dy d2x (13) whence, dividing (12) by the several terms of equality (13), we have

(dy d2z — dz d2y) (§ —x) + (dz d2x — dx d2z) (n—y)

+(dx d2y-dy d2x) (5-2) = 0; (14)

which is the equation to the osculating plane.

344.] The method by which we have deduced this equation is the same as if we had defined the osculating plane to be that in which two consecutive tangents lie, as will be apparent from what follows.

Let the equation to the plane passing through (x, y, z) be

A(x)+B(n− y) + c (5—≈) = 0;

and since it is to be that in which two consecutive tangents lie, whose direction cosines are respectively

A(dx + d2x) + B (dy + d2y) + c (dz + d2z) = 0;

whence, by subtraction,

Ad2x + Bd2y+cd2% = 0;

which two relations between A, B, C are the same as those above marked (10) and (11), whence equality (13) follows, and therefore the equation to the osculating plane is the same.

345] It is manifest from (7) that all straight lines passing through a point of contact, and perpendicular to the tangent line, lie in the normal plane; two of these normal lines have peculiar properties in relation to the osculating plane, viz. that which is perpendicular to it, and that which lies in it, and is therefore the line of intersection of it by the normal plane. The latter is called the principal normal, and the former has the distinctive name of binormal, being, as it is, perpendicular to two consecutive elements of the curve, while all other normals are perpendicular to only one.