= --

{Un−1 + 2 Un-2 + + (n − 1) u1 + nu}, (24)

and is therefore an equation of only (n − 1) dimensions in terms of x, y, and z.

η,

In the equation to the tangent plane, considering έ, 7, to be constant, and the coordinates of a given point through which a series of tangent planes is drawn, x, y, z refer to the points of contact on the surface; hence we have the following theorems:

If through a given point planes are drawn, touching a given surface of the nth order, the points of contact lie on a surface of the (n-1)th order; and therefore

If through a given point planes are drawn touching a surface of the second order, all the points of contact lie in one plane.

In harmony with the nomenclature of Art. 226, the point (§, n,) whence the tangent planes are drawn is called the pole : the surface whose equation is (24) is called the first polar surface, and the surface F(x, y, z) = c is called the base-surface. Also in the same way that the first polar surface is derived from the base-surface and is of the (n-1)th order, so may other and successive polar surfaces with reference to the same pole be derived, and these will be of the (n-2)th, (n-3)th, ...... order. Want of space however precludes me from entering on these subjects, although they are replete with interest.

336.] To find the equations to a normal of a curved surface. A normal is a straight line drawn through any point of a curved surface, and at right angles to the tangent plane at that point.

Let, n, be the current coordinates of the normal, and x, y, z the coordinates to the point where it meets the surface; then, by Art. 333, the direction-cosines of the normal being proportional to (d), its equations are

dr

dx

dr

dz

Also the form of the equations to the normal shews that it is the longest or the shortest line which can be drawn from a point on it to the surface.

If the equation to the surface is given in the explicit form, these equations, by means of Art. 334, become

In these equations, if έ, 7,

(5—2)

(26)

are constant, x, y, z refer to the points on a surface where normals drawn through a given point meet it, and the equations (25) or (26) are those to a curve in space which is the locus of such points of contact.

337.] From (25) it follows, that the equations to a line passing through the origin, and at right angles to the tangent plane, are

×

dr
dx

dr dy

dz

By means of which equations, combined with those to the tangent plane and to the surface, we may determine the equation to the surface, which is the locus of the point of intersection of a tangent plane, with the perpendicular drawn to it from the origin.

338.] Examples illustrative of the preceding Articles.

Ex. 1. The ellipsoid whose equation is

therefore by equation (14) the equation to the tangent plane is

ny ต๊ะ
+ +
a2 62 c2

= 1,

(29)

which is plainly the equation, since the equation to the surface is a homogeneous function of two dimensions.

Also if (έ, n, ) is the pole, the equation to the first polar is

Ex ny (2

a2

+ + = 1;
b2 C2

which is the equation to a plane.

(30)

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

Whence may be found the equation to the surface, which is the locus of the point of intersection of these lines with the tangent planes.

For έ, n, being the same in (29) and (33), we have

=

a2 b2

=

= {2+n2+52;

{¿2+n2+52}2 = a2 §2 + b2 n2 + c2 5 2;

which is the equation to the surface required.

(34)

(35)

and the equations to the line through the origin, and perpen

dicular to the tangent plane, are

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

an2 + b2 + § (§2 + n2 + 52) = 0.

=

Ex. 3. If the equation to the surface is xyz = k3,

(37)

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

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

εx = ny = (z = k (§n¢)*;

(40)

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

339.] If at the point on the surface at which the tangent

lines of equation (11) are drawn, (d), (d), and (dr)

dy

dz

all

vanish, equation (13) is satisfied independently of any relation between da, 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 da, dy and dz, arising out of the equation to the surface. Such we have, if all the differential-coeffi

cients of the second order do not vanish at the point in question, in the differential of (13), and which is also the third term of the expansion of F (x+dx, y+dy, z+dz) in Art. 142, viz.

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

A §2+Bn2+c82 + 2 D n 5 + 2 E § § + 2 r §n = 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, ay2 + bz2 +x (x2 + y2 + z2) = 0.

= 2bz+2xz = 0,

(123) = 2a + 2x = 2a, (dzi) = 2b+2x=2b,

dy2