Page images
PDF
EPUB

To find the equations to the binormal.

Let l, m, n be its direction-angles; then, as it is perpendicular to the osculating plane,

cos /

=

cos m

==

[ocr errors]

dy d2z - dz d2y dz d2x-dx d2z dx d2y - dy d2x

1

{(dy d2z — dz d2y)2 + (dz d2x — dx d2z)2 + (dx d2y — dy d2x)2} ♣

[ocr errors]

;;(15)

The denominator of which last expression may be modified as follows:

(dy d2z— dz d2y)2 + (dz d2x — dx d2z)2 + (dx d2y — dy d2x)2

=

(dx2 + dy2 + dz2) { (d2x)2 + (d2y)2 +(d2z)2}

-(dx d2x + dy d2y+dz d2z)2; (16)

[blocks in formation]

and therefore the right-hand member of (16) becomes

[merged small][merged small][merged small][merged small][merged small][ocr errors][merged small][merged small]

=

=

(21)

dx d2y- dy d2x

whence the equations of the binormal are

dy d2z-dz d2y dz d2x-dxd2z

346.] To find the equations to the principal normal.

[blocks in formation]

then, by reason of its being perpendicular to the tangent line, and of its lying in the osculating plane, we have

Ldx+мdy + N dz = 0,

(23)

L(dyd2z — dzd2y) +м(dzd2x— dxd2z) + N(dxd2y — dyd2x)=0; (24) whence we have

L

dy (dx d'y - dy d2x) — dz (dz d2x-dx d2z)

L

=

dx (dx d'x+dyd2y+dz d2z) — d2x (dx2 + dy2 + dz2)

or,

and since

PRICE, VOL. I.

[ocr errors][merged small][merged small]

(25)

(26)

(27)

3 U

[merged small][merged small][merged small][merged small][merged small][merged small][merged small][subsumed][subsumed][merged small][ocr errors][merged small][subsumed][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small]

Therefore, if λ, μ, v are the direction-angles of the principal normal, we have

[ocr errors][ocr errors][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small]

347.] Examples on the preceding.

Ex. 1. The curve formed by the intersection of an ellipsoid

[merged small][merged small][merged small][merged small][merged small][ocr errors][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][ocr errors][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][ocr errors][merged small][merged small][ocr errors][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small]

(31)

[merged small][merged small][merged small][subsumed][subsumed][ocr errors][subsumed][subsumed][ocr errors][merged small][subsumed][subsumed][ocr errors][ocr errors][merged small]

Ex. 2. The helix; see fig. 125.

Let OA = OB = a be the radius of the base-cylinder of the helix, and AON be the angle between the plane of az and the radius of the cylinder drawn to the point (x, y, z), and whose projection on the plane of xy is on; and let oм = x, MN=Y, NP = 2; and let k be the tangent of the angle at which the thread of the helix is inclined to the plane of xy; so that NP = k x the arc AN; whereby the equations to the curve are x = a cos 4,

y = a sin 4,

z = kap;

(32)

[blocks in formation]

the differentiations being performed on the supposition that is equicrescent; therefore the equations to the tangent are

[blocks in formation]
[ocr errors]

-a (x) sin + a (ny) cos +ka (-2) = 0,

nx-y+ka (5—z) = 0;

(35)

when = n = 0,2; the normal plane therefore cuts the axis of z at a distance from the origin, equal to the z of the helix at which it is drawn.

[blocks in formation]

therefore if λ, μ, v are the direction-angles of the tangent

[merged small][ocr errors][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small]

(36)

(37)

The tangent therefore is always inclined at the same angle to

the axis of z.

or

Hence also the equation to the osculating plane is

ka2 sin(x)-ka2 cos p (n− y) + a2 (5—≈) = 0,

k (y-nx)+a (5—z) = 0.

(38)

Also from (37) and (36), taking s to be equicrescent,

[blocks in formation]

therefore the direction-cosines of the principal normal are, by
reason of (31), cos 4, sin 4, 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 Ax+BY+cz = D; then

A dx + B dy+c dz = 0,

A d2x + в d2y + c d2z = 0,

(40)

▲ d3x + B d3y + c d3z = 0;

whence by cross-multiplication,

dx (d2y d3z — d2zd3y) + dy (d2z d3x — d2x d3z) +dz (d2x d3y — d2y d3x) = 0; (41)

which condition becomes, if z is taken to be an equicrescent

variable,

d2x d3y d2y d3x
= 0;
dz2 dz3 dz2 dz3

(42)

the geometrical meaning of which condition will be explained
hereafter.

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 geodesics 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

[blocks in formation]

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

(હૃ

(ε—x) (dx)

+ (n− y) (dy) + (8 − 2) (dx) = 0; (44)

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

[subsumed][ocr errors][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small]

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

(dr) dy

dr

dx = 0;

dy

(46)

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

[blocks in formation]

if the initial values of x and y are c and b; that is, the line of

greatest slope is a meridianal arc.

« PreviousContinue »