Ex. 3. The Circle, the extremity of a diameter being the pole. In this case the equation is r = 2a cos 0; consequently

If the arc begins at the extremity of the diameter, 0, 0; so

Ex. 4. The whole length of the cardioid, whose equation is r = a(1+ cos 0), is 8a.

Ex. 5. If the equation to the lemniscata is r2 = a2 cos 20, and s the length of a loop,

This last value may be expressed in terms of the Gamma-function.

For let 20 be replaced by1⁄2-20; then

which gives a geometrical interpretation of r

163.] If the equation to the curve is given in terms of r and p, then, by Vol. I, Art. 271, (23),

the integral being taken between limits assigned by the problem.

Ex. 1. To find the length of the involute of the circle, whose equation is 2 a2+p2, between any two given points on it. r2 =

and if s begins at the point where the involute leaves the circle, ro = a; and

p2

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Ex. 2.] It is required to find the whole length of the hypocycloid whose equation is x+y= a; see fig. 10.

The equation in terms of r and p is 3p2 a2 —r2;

=

SECTION 3.-The Rectification of Curves in Space.

164.] The infinitesimal length-element of a curve in space, whether plane or non-plane, has been determined in Art. 341, Vol. I, and is given by the equation

ds=

(dx2 + dy2+dz2)3;

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so that, by integration between the given limits, the length of the arc of the curve may be found.

If the equations to the curve are given in the form x = f(z), y' (z), then

If the equations are given in terms of another variable, say 4, and are of the forms

And if the equations are given in the form

F(x, y, z), =

0, f(x, y, z), = 0,

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ds may be found by means of the total differentials of these two functions.

Ex. 1. To determine the length of the helix between two given points.

Taking the equations to the curve as found in Vol. 1, Art. 347, (32),

If the arc begins at the point where = 0, then

a result which also follows immediately from the geometrical generation of the curve.

Ex. 2. To determine the length of the curve formed by the intersection of two right cylinders, of which one is parabolic and perpendicular to the plane of (y, x), and the other is cycloidal and perpendicular to the plane of (x, z).

Let the equations to the director-curves of the cylinders be y24 cx; and z = a versin-1 + (2 ax − x2)};

.. dy = (-)'dr;

α

Ex. 3.

Prove that the length of the curve of intersection

of the elliptic cylinder, a2 y2+b2x2 = a2 b2, with the sphere

x2+ y2+z2 = a2, is equal to 2πа.

In Vol. XIII of Liouville's Journal is a memoir by M. J. A. Serret which contains a solution, by a process somewhat similar to that given in Art. 160, of the equation

ds2 = dx2 + dy2 + dz2;

and observations on the mode of solution will be found in Art. 8 of Note I, appended to Liouville's Edition of Monge's Application d'Analyse.

165.]. It is frequently more convenient for purposes of rectification, as well as for other problems which will be subsequently investigated, to refer the position of a point in space to polar coordinates, the system of which is as follows.

Let E, fig. 42, be the point in space whose position is to be determined. Take a fixed point o for origin, and through it draw a plane which I will call the fundamental plane, and, to fix our thoughts, will assume to be horizontal. Through o draw oz perpendicular to, and also any line or in, the fundamental plane. Join OE; OE is the radius vector of E and is denoted by r, and the angle Eoz, at which OE is inclined to oz, = 0. Let a plane be drawn through oz and OE, and let oN be the line in which this plane cuts the fundamental plane; then as these two planes are perpendicular to each other, ON is the projection of OE on the fundamental plane. ON is called the curtate radius vector, and is denoted by p; so that ON = p = r sin 0. Also let another plane be drawn through oz and ox, and let & be the angle between it and the plane zOE; or, what is the same thing, let be the angle xON; then these quantities r, 0, & completely determine the position of E, and are consequently called the polar coordinates of E; so that when they are given, the place of E is also given. This system is equivalent to a spherical system of coordinates, and is that by which position on the surface of the earth is usually determined. o is the centre of the earth, oz is the polar axis, the fundamental plane is the plane of the equator, the plane zor is that from which longitude is measured, and in the English system passes through Greenwich Observatory; thus is the longitude and is the co-latitude, and when they are given, position on the surface of a sphere is given.

If r is of given length, and ℗ and vary, the extremity of r describes a sphere. If is constant and varies, r describes a right conical surface, which intersects a spherical surface along a parallel of latitude. If is constant and 0 varies, r describes a

meridian plane. It is evident that the sphere, the cone, and the meridian plane intersect orthogonally at every common point.

To compare this system with that of rectangular coordinates, let the fundamental plane be that of (x, y); and in it let oy be drawn perpendicular to ox: let E be (x,y,z); then, since Oм=x, MN = y, NP = 2; OE = r, ON = p, 20E = 0, xON= 4, therefore

and by means of these equivalents, an equation may be transformed from one system to another.

In reference to the problem of rectification, let the preceding equivalents for x, y, z be differentiated; then we have

dx = sin 0 cos o dr+r cos 0 cos de ―r sin 0 sin o do,

dy = sin 0 sin & dr + y cos 0 sin 4 de+r sin 0 cos & do,
dz = cos 0 dr -r sin 0 do ;

..

ds=

(dx2 + dy2 + dz2) ś

= {dr2+r2 do2 + r2 (sin 0)2 dp2}1.

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This last expression however may be determined as follows. As ds is the distance between two points infinitesimally near to one another in space, let E and H be the two points, see fig. 42, where E is (r, 0, 4), and H is (r+ dr, 0 + d0, $+dp). Let EJ dr; and, being constant, let the radius vector OE revolve through an infinitesimal angle de in the meridian plane thus determined, so that E describes the small circular arc EF which rde; and finally let the meridian plane revolve through an infinitesimal angle do, whereby, 0 being constant, E describes the small circular arc EI which = NQ = pdp r sin 0 do. Now these three lines EF, EI, EJ, which meet at E, form such a system that each is at

=

right angles to the other two; they may consequently be considered a system of rectangular coordinates, which assigns the coordinates of the point H; and consequently, as EH = ds,

Ex. 1.

ds = {dr2 + r2 d02 +r2 (sin 0)2 dp2)1.

Determine the length of the curve of intersection of the right cone and the sphere whose equations are respectively