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The equation therefore to the locus of the point of intersection of (36) with (35) is

arf + b{* + t(t* + r? + P) = 0. (37)

Ex. 3. If the equation to the surface is xyz = k3, idr\ k3 /dr\ k3 /dv\ k3

uw=y y=T; (38>

and therefore the equation to the tangent plane is

13 13 It

(£-*) —H in-y)— + (C-z)— = 0,


x + l + iz = S- <39)

And the intercepts of the coordinate axes by the tangent plane are, according to the notation of Art. 219,

fo = 3 r, rjn = 3y, Co = 3z;

.-. to*io(o = 27xyz,
= 27 k3,

that is, the volume of the pyramid contained between the tangent plane and the coordinate planes is constant.

The equations to the line through the origin, and perpendicular to the tangent plane, are

fj? = ny = Cz = *(f,0*; (40)

therefore the equation to the locus of the point of intersection of (40) with (39) is

fa + if + f1 = 3*(f,f)*- (41)

339.3 ^ at tne Pomt on the surface at which the tangent

lines of equation (11) are drawn, an<* (5^) *^

vanish, equation (13) is satisfied independently of any relation between dx, dy and dz, and therefore does not give an equation whereby to eliminate them; in fact the direction-cosines of the normal at the point are indeterminate, and the tangent plane has no definite position. At such a point there will be a locus of tangent planes, to determine which we must seek for some other relation between dx, dy and dz, arising out of the equation to the surface. Such we have, if all the difFerential-coeffi

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an equation of a cone of the second degree, shewing therefore that the locus of the tangent lines is not a plane, but a cone of the second order.

Changing the origin to the point under consideration, the equation assumes the form

Afa + B^ + cf3 + 2D,f-l-2Eff + 2r{r, = 0, (43)

and the vertex of the cone is at the point of contact; and it may happen that the coefficients have such relations that the equation is decomposable into two factors of the first degree, in which case it will represent two planes.

Ex. 1. Determine the nature of the point at the origin of the surface whose equation is, ay1 + bz2 + x(x2 + y2 + z2) = 0.

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which is satisfied only by y = 0, z = 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

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

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.

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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 formula? 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 f, t), ( 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 + &.X, y + Ay, z + &z); then the equations to the line are

t-x __ v-y _ f-~ _ r

Al' Ay AZ A«'

where r is the distance between the two points (x, y, z) and (f • Vi ()• When these two points become infinitesimally near to one another, the line becomes a tangent, and its equations become ,

1^ = 1^1= Lli = 2L. (2)

dx dy dz ds'

where ds = (dx* -f dy2 + dz2)b,

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 A, fi, v are the direction-angles of the tangent,

dx dy dz ,_.

cos A = —, cos u = cos v = -r. (3) ds ds ds

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

forms f(x,z) = 0, <t>(y,z) = 0,

dx dy

-r- and ^ 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,

n (x, y, z) = 0, F2 (x, y,z) = 0; (4)

then, since

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\dx'\dy' \dyl\dx>

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

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