The inclination of a straight lime 1 to each of the axes :

ta cos 1,8 =

(1 + a2 + b)

+5 cos l3y=

(1 + a2 +69)1'

+1 cos 1,7 =

(1 + a® +62) (cos 1,ix) + (cos 1,y) + (cos 1,x) = 1. If lyxy, &c. denote the angles which I makes with the planes xy, &c. then

(sin lyxy)+ (sin lyxz)' + (sin lyx)* =1. The mutual inclination of two lines, l, le: cos ligle = cos 119X.cos lygx + cos lı,y.cos l2,y + cos 11,7.cos l25%.

1+aja, +b, bę +

(1 + a,* +6,*)(1 + a, + 6,397 The equations to a line in terms of the angles it makes with the axes are cos lyx

cos lny

%+ ß. cos 1,7

cos 1,2 (G. G. A. Ch. xvi; H. A. G. 229–51; Biot, 54-69.) (26.) The plane. The most general form of the equation to a plane is

A x + By + Cx+D=0, or

x= A,X + B y + C1. The equations to the traces of the given plane on the planes of xy, xz, yz, respectively are AX + By+D=0, A x + Cx+ D=0, By + Cx+D=0.

If a, b, c, are the distances from the origin at which the plane cuts the axes of x, y, %, respectively, the equation becomes

y =

If
p is the perpendicular on the plane from the origin, then
prix cos poil + y cos poy + x cos p.č.

Z

=

The equation to a plane passing through the point (19913x_), and parallel to the plane Ax + By + Cx+D=0, is

A(x — «) + B(y-yl) + C(x– x)=0.
The co-ordinates of the intersection of a straight line
X = ax + a,

y=bx + ß and the plane Ax +By+Cx+D=0, are

a(Aa + B6+C) – a(Aa + BB+ D)

Aa + Bb + c
B(Aa + Bb + C) – b(Aa + BB+ D)
y=

Aa + Bb +C
Aa + BB+D

Aa + Bb C
When the line and plane are parallel, then

Aa + Bb+C=0.
The equations to a perpendicular p from the point
(12919%) on the plane, are
A

B
X – X1=

y-y=

-;
Axı + Byı + C % + D
p=+

(A + Bo + C2) The equation to a plane drawn through a point (X1,91,1) perpendicular to a given line, is

a (x — «1) +b(y-yı) +%—%=0. The angle contained between the line l, and the plane P;

Aa + Bb+C sin 1,P=

(1 + a2 + b)(A + B? + C) (27.) The inclination of a plane P to the co-ordinate planes :

+C cos Pwy=

(A + B2 + C'?),

A1

Br

cos P,yx=

(A® + B? + C)} (cos Py) + (cos P,xz)2 + (cos P,yx)?=1. The mutual inclination of two planes P1, P2; cos P1,P,=cos Powy.cos P29XY + cos P 19xx. cos P 298 %

+ cos P 1,7x.cos P2Yz.

A,A, + B, B, +C,C,

(A,? + B,” + C*)(A,? + B +0,2)
If the planes are parallel, then

B C
A2

C, (H. A. G. 252—74; Biot, 70–85; G. G. A. Ch. xvii.) THE ORTHOGONAL PROJECTION OF PLANE FIGURES. (28.) Let A represent the area of a plane figure, Ary, Ag, Ayes the areas of its projections on the planes xy, xx, yx respectively, then

4., +42 +4,2 = A.
If A, Ays, be projected on the plane yım, of some
system, then

Ay,;, = Ay cos X19X + A, COS XY + A,, cos X1,2,
If A1, A2, A3, &c. are any areas in different planes,
X, Y, Z, the sums of their projections on yx, xx, xy;
X, Y, Z,

Y1X1, X 1% X 19ı : then X,= X cos X19X + Y cos X1,Y + 2 cos X1,%,

Y = X cos Y 1X + Y cos Yı,y + Z cos Y19%,
2,=Xcos %1X + Y cos x1,4 + 2 cos %19%.

X2 + Y? + Z2 = X + Y + 2%: If the triangle formed by joining any three points in space be represented by a, and the perpendicular on the plane from the origin, by p, the equation to the plane passing through the three points is

ays. + arzy + aryX=ap.

(H. A. G. 275–90; G. G. A. p. 327–32.)

new

.

.

.

.

OBLIQUE CO-ORDINATES. (29.) The equations to the line ( referred to oblique coordinates are

% + a,
y=

%+ß.

sin 1,298 sin brzyd y

The distance of the point (x,y,z) from the origin

= x2 + y + x2 + 2(xy cos x,y + x x cos x,x+yx cos y,x). The distance between the two points (27,41,71), (X2392,%,)

=(2, - x) + (,- y.) + (2,-2)
+2{(x - 2)(y1 - y.) cos x,y + (x1—,)(x1 — %,) cos X,X

+ (yı- y2)(x1 - x) cos y,x}.
The inclination of two lines
sw=a7x + ai

sx=0,7 + a, (1.),

(12): ty=b3 +8. ly=b2x + ß

sin lyx sin x,y% sin 11,4% sin a',y%

sin 10,0 %

sin lXY

cos 11%, cosly +

sin %,XY sin lyx x

sin llwy

cos 1.,%; cosloy + sin Y,& %

sin %9X Y

cos 1 +

dyda+b7b2+1+(a,b2 + a2b)cos x,y+(a + a2)cos x,x+(62 +62)cosy,

{a, + b +1+2 (a, b, cos #3y + a, cos x,x+b, cos y,x)} x {a, + b3 +1+2 (a,b,cos x,y + a, cos x,x+ b, cosy,x)}

The inclination of a plane P to each of the co-ordinate planes ; let p be the perpendicular from the origin, then the equation to the plane is

p=X COS Px + y cosp,y + % CoS pox,
= AX + By + Cx.

1
COS Y 7,2 %

cos Y ZNY
A+
B +

C,
ços P,yx=
sin x,y%
sin y,xx

sin

% XY
1
COS X 2,9%

COS X 2,8 Y
B+

A+
cos P,xx=

sin Y,8 %
1

COS XY,Y% COS X Y,8 %
C +
· A+

B.
cos Pwy=

sin y,XX sin x,xy

C,

sin x,y%

sin 2,XY

sin 2,4%

B + sin Y,8 %

-24.5.

cos P,7% cos P,xx cos P,wy
A +

C=1.

sin %,xy Let / xy, = a,

xyyyx=ß,

2 X ,yx = y; sin x,yx=d, sin y,xx = e,

sin x,xy=f: then A B2 C2 AB

AC

BC + + 2

COS

cos ß + do

f2 de df ef The inclination of two planes

Ax+By+Ciz=pı, (P)

A2X + B,y + C,x=po; (P2) - cos P1,P,= A,A,, B,B2, CC, A,B,+A,B A,C,+ A,C, B,C,+B,C

+ + ď e

de
df

ef
14, B, C, A,B,

B,C
+
+

2 COS Y + cosB +
de
df

ef
(A, B,
B, C,

A.C.

B,C,
+
+

cos y +

cos B+ e? f2 de

df

ef (Whewell, Camb. Phil. Trans. V. 2, Pt. 1; H. A. G. Ch. ix, x.)

1

cos

cos y

Х

d

THE TRANSFORMATION OF CO-ORDINATES.
(30.) Let x, y, z, be the original, X1, 91, X1, the new co-ordi-
nates, then

1

(x, sin x 1,4x + yi sin Y14x + x, sin %17%),
sin 2,4%

1
y= (x, sin & 7xx + y, sin Yı,0% + x, sin %1,0%),
sin y,# %
1

(x, sin xy + y, sin yıxy + x, sin x wy).

X=

sin my

If the primitive axes are rectangular, and the new ones
oblique, then

X = X, COS X 1920 + yu cos Y198 + %1
y=&, COS X 1,4 + Yi cos Yı,y + %1

COS %1,9,

(1) %= X 1 COS X 1,7 + Y, cos Y 1,7 + %, COS %19%: