The equation to the paraboloid is a2 + y2=4mx;

(H. A. G. 347-58; G. G. A. 194-9; Biot, 312-21.)

(40.) Surfaces which have a centre. If the origin is transferred to the centre, the general equation becomes

Ax2 + By2 + C≈2+2(Ã ̧y≈+ B1x≈ + C1xy) + D=0.

If the axes coincide with any system of conjugate diameters, the equation is reduced to

a, b, c being the semi-axes; in which case, the co-ordinates are rectangular.

Let a1, b1, c1 be any system of conjugate diameters, then,

by

C1

If the surface be referred to the diameters a1, b1, c1, then (a1b1. sin x,y)2+(α1c1 . sin x,≈)2+(b1c1. sin y,≈)2=a2b2+a2c2+b2c°. and a2bc2= a,b,c,{1-2 (cos x,y. cos x,x.cos y,x)

− (cos x,y)2 — (cos x,≈)2 — (cos y,≈)2}.

(41.) The species of the class of surfaces defined by the equation

depend on the signs of a2, b2, and c2; the equation to

In the ellipsoid, the three traces, or principal sections are

If two of the quantities a2, b2, c2 are equal, the surface

is a spheroid, if they are all equal, a sphere.

In the hyperboloid of one sheet, the principal sections are

is an asymptote to the hyperboloid.

In the hyperboloid of two sheets the principal sections are

(H. A G. 359-81; G. G. 4. 200-24; Biot, 322-30.)

(42.) Surfaces which have not a centre. If the origin is at the vertex, and one of the co-ordinate axes, as x, coincides with the axis of the surface, the general form of the equation is

My+N+Px=0.

This class consists of two species,

the elliptic paraboloid, ny2+ m2 = 4mnx,

the hyperbolic paraboloid, ny3 — m22 = 4mnx. The principal sections of the elliptic paraboloid are a parabola, y2 = 4mx,

This surface will be generated by the parabola x2=4nx moving parallel to itself so that its vertex may describe the parabola y2 = 4mx.

The principal sections of the hyperbolic paraboloid are

two straight lines, ny2 — mx2=0.

All sections parallel to xy and xx are parabolas, and parallel to yx, hyperbolas.

The two planes defined by the equation ny2 - m22=0 are asymptotes to the surface.

This surface will be generated by the parabola 2= — 4nx moving parallel to itself, so that its vertex may describe the parabola y=4mx.

(H. A. G. 382-95; G. G. A. 225-31; Biot, 331-4.) THE INTERSECTION OF A SURFACE OF THE SECOND ORDER AND A PLANE.

(43.) Let the surface be referred to its centre and axes, the equation is Ax2 + By2+Cx2 + D=0,

The equation to the line of intersection is

{4 (cos() + B (sin()2} x2 + 2 (A−B) sin d.cos .cos x.xy

+ {(cos x)2. A (sin q)2 + B (cos ()2 + C (sin x)2} y2+D=0;

the equation to a curve of the second order, which will be an ellipse, a parabola, or an hyperbola, according as

AB (cos x)2 + C(sin x)2 { B (sin p)2 + A (cos q)2 } >, =, or <0. In the ellipsoid, every section is an ellipse, or one of its varieties.

In the hyperboloids, the section may be either an ellipse, a parabola, or an hyperbola.

(44.) If the surface have not a centre, the equation is

My+N+Px=0.

The equation to the line of intersection is

{M (cos .cos x)2 + N (sin x)2} y2 + N (sin ()2x2

-2M sin p.coso.cos x.xy + P cosp.cos x.y + Pcos .x=0; the equation to a curve of the second order, which will be an ellipse, a parabola, or an hyperbola according as

MN (sin p.sin x)3>, =, or<0.

In the elliptic paraboloid the section is an ellipse, and in the hyperbolic paraboloid an hyperbola, except when

or x=0, in which case the section is a parabola.

=0,

(45.) If the surface has a centre, the section will be a circle, when

2

A (sino)2+B (cos p)2=C' (sin x)2+ (cos x) { 4 (sin ()2+B (cos ()3}, 2 (4-B) sino.cos.cos x=0.

These equations will be fulfilled if the plane is

B

perpendicular to yx, and tan x= ± (3–4)',

A

A-B

+ (B-C)

= ±

of which three quantities only one is possible.

If the quantities a, b, c, be arranged in the order of magnitude, the section intersects the mean axis 2b in the ellipsoid, and hyperboloid of two sheets, and the greatest axis 2a in the hyperboloid of one sheet.

The inclination of the section to the plane ay is

(46.) If the surface has not a centre, the section will be a

circle, when

M (cosX.cos¢) + N{ (sinx) − (sin¢)}=0,

THE TANGENT PLANE.

(47.) If the equation to the surface is

Ax2 + By2+ C≈2 +2(4x+ B2y + C2≈) + D=0;

the equation a plane touching the surface at the point (≈111) is (Ax1+A ̧)x+(By1+B2)y+(C≈1+C2)≈+A ̧¤1+B2Y1+C2≈1+D=0.

2

If a · x1 = a (≈ — ≈1), y—y1=b (≈ —1) are the equations to a line passing through two points (x,y,z), (~1,Y1,≈1), the point (~2,2,*) in which the plane passing through these points touches

the surface may be determined from the equations

2

Ax2 + By22 + C≈22 + 2 (A ̧‰1⁄2 + B2Y1⁄2 + C2≈2) + D=0,

2

2

(A x2 + A2) x1 + (By1⁄2 + B2) y1 + (C≈2 + C2)≈1

X1

2