These values of x, y, z being substituted in (1), the resulting equation will express the nature of the curve required,

Note. In the actual application of these formulas, it will be found convenient to denote the angles involved in them by single letters. We shall assume, therefore,

▲ x'y', xy=0,

Lt, x, =0,

4 t, x′, ・ =4.

322. PROP. To pass from rectangular, to polar, coordinates.

Let P be any point in space, referred to A as the pole, the radius vector (r) being AP.

Then the simplest form of the polar equation to P, will be that which involves r and the angles r, x; r,y; r, z.

The angles, however, most commonly employed, are those which the projection of r upon xy forms with r, and with the axis of x.

Hence, if conformably to the notation used in Chap. viii. r denote the projected radius vector, the polar co-ordinates will

be

r, and the angles r,r; r,,x,

2

Let x, y, z be the rectangular co-ordinates of P, namely,

AM, MN, NP, (fig. 76.), and join AN, AP.

Then it is evident, that AN=r, and that AMN and PAN are right angles.

Hence, AMAN cos MAN= AP cos PAN cos MAN,

which values of x, y, z, when substituted in any equation between these quantities, will furnish the polar equation required.

If the pole be transferred to any point S (a, ß, y), we have only to annex a, ẞ and y respectively to the values of x, y and z just deduced.

Observation. If the angle ra, x be supposed to vary from 0 to 360o, and the angle rz, r from 0 to 90°, then the position of each point in space will be completely determined by the signs of the trigonometrical functions of these angles. The sign, therefore, of the radius vector must always, as in Part I, be considered positive.

SECTION II.

ON THE SPHERE, CONE, AND CYLINDER; AND ON

SURFACES OF REVOLUTION,

CHAP. I.

ON THE SPHERE.

323. PROP. 1. To find the equation to the sphere.

The equation sought will be obtained by expressing analytically that the distance between the centre and any point on the surface of the sphere, is invariable.

Let a sphere, whose radius is (r), be referred to any system of oblique axes; suppose x', y', z' to be the co-ordinates of the centre, and x, y, z those of any point on the surface.

Then, the distance between the two points (x', y', z′) and (x, y, z) is expressed by the equation

(x − x')2+(y—y')2+(z — z′)2

+2(x − x') ( y − y') cos x, y +2 (x − x′) (z — z′) cos x, z

+2(y — y') (z — z′) cos y, z= r2,

which is the equation required.

COR. When the origin is at the centre, then x, y, z′ are each =0, and the equation becomes

x2+y2+z2+2xy cos x,y+2xz cos x, z+2yz cos y, z = r2.

324. PROP. 2. To find the various forms which the equation to the sphere assumes, when the axes are supposed to be rectangular.

Since cos x, y, cos x, z, and cos y, z each =0, the general equation becomes

(x − x')2+(y−y')2 + (z − z')2 = r2 . . . . . .(1).

Let the origin be upon the sphere.

12

12

12

2

Then, since x2+y"+z"2=r2, equation (1) becomes

x2-2xx' + y2-2yy' + z2-2zz'=0......(2).

Let the origin be upon one of the co-ordinate planes, upon that of xy, for example.

Then, since z=0, equation (1) becomes

(x − x')2 + (y−y')2 + z2 = r2 . . . . . . .

(3).

Let the origin be upon one of the axes, upon that of z, for example.

Then, since y' and ' each =0, equation (1) becomes

(x − x')2 + y2+z2 = r2..

(4).

When the origin is supposed to be upon either of the planes xz, yz, or upon either of the axes of y, z, the form of the equation will be similar to that of (3) or of (4).

Let the origin be at the centre.

Then, since x', y and z' each =0, equation (1) becomes

325. PROP. 3. To find the intersection of the sphere with each of the co-ordinate planes.

Let the sphere intersect the plane of xy.

Then, since z=0, equation (1), in the last article, becomes

'(x − x')2 + (y—y')2+z'2=r3 ;

2

•'• (x−x')2+(y− y')2 = r2 — z2,

which is the equation to a circle, the co-ordinates of whose centre are and y', and whose radius = √(r-2'3).

In like manner it may be shewn that the sections of the sphere, made by the planes of xz and yz, are circles.

326, PROP. 4. To find the section of the sphere made by any plane whatever.

Let the sphere be referred to a system of rectangular coordinates originating at the centre; then, its equation is

The intersecting plane being supposed to pass through a point (a, b, c), the nature of the section will be determined by substituting for x, y, z, in (1), the following values (321)

This operation being executed, and the resulting expression reduced, we have

y2+x22+2 {(a sin -b cos p) cose+c sin 0} y'+2(acos &+bsin () x′