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The equation to the paraboloid is mo? + yo=4mz;

z? spheroid

a 62

in + y hyperboloid ..

-1.

a? 62 (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

A x2 + By® + Cx2 + 2(4,4% + Bxx + Cay) +D=0.

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

Ax+ By® + Cx +D=0,

у*

c a, b, c being the semi-axes; in which case, the co-ordinates are rectangular. Let an, bz, C, be any system of conjugate diameters, then,

a + b + c = a + b2 + co. If the surface be referred to the diameters a, b, c, then (a,b,.sin x,y) +(a,C,. sin 8,x)? +(67C7. sin y,x)*=aRb + a co+boco. and a’b*c= a,b,c,'{1–2 (cos X,Y.COS X ,7.cos y,x)

- (cos x,y)2 – (cos x,x)* — (cos y,x);}. (41.) The species of the class of surfaces defined by the equation

y? - g?

a 72 c? depend on the signs of a’, b, and cʻ; the equation to the ellipsoid is

:1; 62 c

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=l,

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the hyperboloid of one sheet,

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:1;

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the hyperboloid of two sheets,

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In the ellipsoid, the three traces, or principal sections are ellipses, their equations are

yo

=l, the trace on ym;

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If two of the quantities , b, c are equal, the surface is a spheroid, if they are all equal, a sphere. In the hyperboloid of one sheet, the principal sections are

x2

y an ellipse,

62

=1,

an hyperbola,

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=1.

+ a?

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The conical surface the equation to which is

y

62 is an asymptote to the hyperboloid. In the hyperboloid of two sheets the principal sections are an hyperbola,

= 1. a 62

2

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=1,

+

у? an imaginary curve,

- 1.

32
The plane yx does not meet the surface.
The equation to the asymptotic cone is

20

= 0. a 62 c? (H. AG. 359—81; G. G. A. 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° + Ngo+Pa=0.
This class consists of two species,

the elliptic paraboloid, nyo + mx* = 4mn x,

the hyperbolic paraboloid, nyo mx'=4mnx, The principal sections of the elliptic paraboloid are

a parabola, y'= 4mx,

x"=4nx,

a point, ny? + mx=0. This surface will be generated by the parabola z=4nx moving parallel to itself so that its vertex may describe the parabola = 4mx. The principal sections of the hyperbolic paraboloid are a parabola, = 4mx,

= -4nx, two straight lines, nyo mzé=0. All sections parallel to xy and xx are parabolas, and parallel to yx, hyperbolas.

The two planes defined by the equation nya mx'=0 are asymptotes to the surface.

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

H. A. G. 382–95; G. G. A. 225-31; Biot, 331-4.)

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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

A x2 + By? + Cxo + D=0, The equation to the line of intersection is {A (cosp)* + B (sing)?} #2 + 2 (A– B) sin 0.cos p.cos X•XY

+{(cos x)*. A (sin o)* + B (cos )+ C (sin x)"} y + 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)* + C(sin x)* {B (sin o)* + A (cos p)"}>, =, 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 + Nx + Px=0.
The equation to the line of intersection is

{M (cos p.cos x) + N (sin x)?} y + N (sin )* x2 – 2M sin O.coso.cos X-XY + P cosp.cos Xy + P cos 0.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)'>, =, or <0. In the elliptic paraboloid the section is an ellipse, and in the hyperbolic paraboloid an hyperbola, except when p=0, or x=0, in which case the section is a parabola. (45.) If the surface has a centre, the section will be a circle, when A (sin p)*+B(coso)=C(sin x)* +(cosx)^{A(sin p)* +B(coso)"},

2(A - B) sino.cos p.cos x=0. These equations will be fulfilled if the plane is

B-A
perpendicular to yx, and tan x= +

A-B
X%,

B-C

A-C ily,

+

C-B of which three quantities only one is possible.

If the quantities ař, bo, cʻ, be arranged in the order of magnitude, the section intersects the mean axis 26 in the ellipsoid, and hyperboloid of two sheets, and the greatest axis 2a in the hyperboloid of one sheet.

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The inclination of the section to the plane xy

is

a? in the ellipsoid,

+

62

с

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-1 tan

2

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ca 7? hyperboloid of one sheet, tan - 1 +

b + c?

a’ +62 two sheets, tan +

с

a

(46.) If the surface has not a centre, the section will be a
circle, when
M (cosx.cos )? + N{(sinx)*-(sino)°}=0,

2 M sin q.cos p.cos x=0.
These conditions will be fulfilled if the plane is

N1
perpendicular to wz, and sin x= +

vy, and sin ø= + (%'

(H. A. G. 396—401; Biot, 332–4.)

THE TANGENT PLANE.

(47.) If the equation to the surface is

Ax? + By® + Cxo + 2 (A2W + B,y + C2%) + D=0; the equation a plane touching the surface at the point (X1,91%) is (Ax2 + A2)x+(By,+B.)y+(Cx+C2)x+ A2X2 +B2%. +Com+D=0.

If æ— &=a(z— %,), y-y=b(7—%,) are the equations to a line passing through two points («,y,), (x129137,), the point (0,99,9%,) in which the plane passing through these points touches the surface may be determined from the equations Ax2 + Byz® + Cx + 2(4,1, + B,Y. + Cox) + D=0, (AX, + A,) &, +(Byg + B.) 4. + (Cx, + C2)%, + A9X, + B,y2 + C2%, +D=0,

AaX, + Bby2 + Cx,=0.

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