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(cos x7,x)+ (cos X 1,4)+ (cos 21,2)=1,
(cos Y 13W) + (cos Yusy) + (cos Yı,x)=1,

(cos x7,x)+ (cOS X1,Y)2 + (cos %1,2)=1.
If both systems are rectangular, then
COS X 1910. Cos Y1, + cos X164 . cos Y1y+cos 87,3.COS Yı,x=0,
cos & Jyll.cos % + cos Y.cos %1,4 + cos x 75%. Cos %1,x=0,

cos Y198. COS Zpgif + cos Yuy.cos %1,9 + cos y ,%. COS %1,%=0. (31.) Let t represent the trace of the plane ay on xyı, then if both systems are rectangular, the position of the new axes may be determined from the three angles

t,x=p, t,x, = y, XY,X,Y,=x: The formulæ (1) become respectively x=x, (cosp.cosy +sin p.cosy.cos x)

+y. (- cos p.sin ys + sin q.cosy.cos x - x, sin p.sin x, y=&(sin Ø.cos y cos Q.sin f.cos x)

y (sin p.sin y + cos O.cos ys.cos x) +%, cos .sin x,

sin ys.sin x + y, cos y sin x + %1 If t be supposed to coincide with X7, these formulæ become

=X, cos 0 + y, sin p.cos X,
y=x, sin o-y, cosq.cos x+x, cos p.sin x, (2)
x=y1

sin x + %1 COS X:
If the origin alone be changed, then
X=X1 + , y=y. +b,

%=% + c. If the position of the origin, as well as the direction of the axes be changed, then a, b, c, the co-ordinates of the new origin must be respectively added to the above values of x, y, and z. (32.) The equation to any surface being given, to find the equation to the curve formed by its intersection with a plane: let the equation to the surface be

f (x,y,z)=0, and the plane of x,y, the intersecting plane, then the trace of

= 1

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X =

sin p.cos Xi

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xzy, on xy being assumed the axis of x, and a line perpendicular to x, in the plane x,y, the axis of Yı, (2) become X = X, COS 0+

(3) y=X1 sin - Y, cos p.cos x;

%=Yı sin x These values of x, y, z being substituted in the equation to the surface will give the equation to the curve required. (33.) Transformation of rectangular to polar co-ordinates.

Let u be the radius vector, Ur, its projection on the plane wy, and a, b, c the co-ordinates of the pole, then

w=U COS UzyąU.COS U rygd + a,
y=U COS UryU.

.sin

Uxyga +b, %= u sin

Ur The radius vector must always be considered positive. (H. A G. Ch. xi; G. G. A. Ch. xviii; Biot, 97–103.)

THE SPHERE. (34.) Let a, b, y, be the co-ordinates of the centre; and r the radius; the most general form of the equation is 2{(x—a)(y-B)cos x,y+(2-a)(z-7) cos & ,x+(y-B)(z-7)cosy,r}

+ (x − a)+ (y - 3)2 + (x-7)*=p2. (1)
If the origin is at the centre, the equation becomes
x? + y + x + 2(xy cos x,y + xz cos x,x+yx cos y,x)=re.
If the axes are rectangular, (1) becomes

(x — a)+ (y - 3)2 + (x-7)= ».
If the origin is on the surface of the sphere, then

x? + yo + x2 – 2 (aw +By+yx)=0.
If the origin is at the centre, then

co +v+cosro. The equation to a plane touching the sphere at the point (x1941,71) is (2-a)(x,- a) + (y - 3)(y-2) + (x-7)(x-7)=7%. If the origin is at the centre, this equation becomes

X ®, + yy, +xz,=r?. (H. A. G. 3239.)

THE CYLINDER.

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(35.) The general equation to a cylindrical surface, the directrix of which is in the plane xy, is

f (x -- az, y-bx)=0. The equation to an oblique cylinder on a circular base, in the plane xy, the origin being in the circumference of the base, is

(x ax)2 + (y-bx) = 2r (x az). Every section of this cylinder made by a plane inclined to its axis is an ellipse, or a circle.

(H. A. G. 331–4.)

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(36.) Let a, ß, y, be the co-ordinates of the vertex, and let the directrix be in the plane xy, the general equation to a conical surface is

y-3

= 0.

13The equation to a right cone on a circular base in the plane

72
+

X-7
If the origin is at the centre of the base, the equation

72
becomes

-?

2

2

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a

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2

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The equation to a right cone, of which the base is the ellipse

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+

62

y The equation to an oblique cone on a circular base, the origin being in the circumference of the base, is

(ax-7X) + (Bx-yy)' = 21.(x-7)(ax-gw). If the origin is at the centre of the base, the equation becomes (ax-x) + (Bx-yy) =ml (x-7).

A

(37.) The equation to the intersection of the cone

22

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with a plane passing through the origin, perpendicular to the plane wz, and inclined to the plane wy at an angle

X

is {(y cos x)' – (r sin x)"} y;? + gčx + 2yp sin X•Y, page=0.

If tan x<, the plane is inclined to the side of the cone, and cuts only one sheet of the surface; in this case, the curve is an ellipse.

If tan x=, the plane is parallel to the side of the cone, and the curve is a parabola.

If tan x>, the plane cuts both sheets of the surface, and the curve is an hyperbola.

The equation to the intersection of the oblique cone

g? (x2 + y) +(a* — Joe) xi — 2aywx+ 2aréz + g&pl=0, and a plane situated as above is (y cosx-asin x) – (r sin x)*} y;° +7°x, * +2 aré cos x.yy-gono=0.

The curve is an ellipse, a parabola, or an hyperbola, according as tan x<, =, or>

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2 ay

If the section is a circle, tan x=

a-gi- que (H. A. G. 335–46; G. G. A. Ch. xv; Biot, Ch. vi.)

SURFACES OF THE SECOND ORDER. (38.) The general form of the equation to a surface of the second order is Ax? +By+ +Cxo +2(A2yx+B,2x+Cqay+42x+B2y+C2%)+D=0.

If the axes are parallel to any system of conjugate diametral planes, the equation becomes

Aw* + By° + Cxo + 2 (A,« + B2y + C2x) + D=0. (1)

A straight line cannot intersect a surface of the second order in more than two points.

AA

y=nz, is

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The equation to a diametral plane, which is the locus of the points of bisection of all chords parallel to the line

X=mx, (Am + Cin + B) x + (C7m + Bn+ A) y +(B,m + Ayn + C)

+ A,m + B2n + C,=0. The number of systems of conjugate diametral planes is unlimited; of these, however, only one system can be rectangular. The co-ordinates of the centre are A, (4- BC) + B, (CC, -AB.) + C,(BB,- A,C)

ABC +2 A,B,C, - AA BBP CC; B,(B,' AC) + C2(AA- B,C) + A, (CC, - A,B,) y=

ABC + 2 A,B,C, - AA BB,- CC,
C,(C' AB) + A, (BB, - A,C)-B,(AA, -B,C)

ABC + 2 A,B,C, - AA BB-CC
If the surface is finite, then

A,' BC < 0, B - AC < 0, 0, - AB<0,

AA + BB+ CC;? ABC - 2AB,C, <0.
If the surface has not a centre, then

AA? + BB + CC = ABC +2A,B,C,. (39.) Surfaces of revolution. If the axis of revolution coincides with one of the co-ordinate axes, as a, the general equation to a surface of revolution is

y2 + x=f(w). The conditions which determine a surface of the second order to be a surface of revolution are

AC(C A) B.(C – 4,1)=0,

B,C,(C-B) - A(C2-BI)=0: also A, B, C, must all have the same sign. The equations to the axis of revolution are

Cu

C

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A

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