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The inclination of a straight line l to each of the axes:

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(cos l,x)2 + (cos l,y)2 + (cos l,x)2 = 1.

If l,xy, &c. denote the angles which

planes xy, &c. then

makes with the

(sin l,xy)+(sin l,xx)2 + (sin l,yx)2 = 1.

The mutual inclination of two lines, l1, l:

cos l12 = cos l,x.cos l,x + cos l,y.cos l2y+ cos l1,. cos l,.

lle

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The equations to a line in terms of the angles it makes with the axes are

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(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

Ax+By+C≈ + D=0, or

≈=A1x + B1y + C1·

The equations to the traces of the given plane on the planes of xy, xz, yx, respectively are

Ax+By+D=0,

Ax+Cx+D=0,

By+C+D=0.

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

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is the perpendicular on the plane from the origin, then

p = x cos p,x + y cos p,y + ≈ cos p,z.
Z

The equation to a plane passing through the point (X1,Y1981), and parallel to the plane Ax + By +C≈+D=0, is

A (x − x ̧) + B(y — y1) + C(≈ — ≈1)=0.

The co-ordinates of the intersection of a straight line
x=az+a, y=bx+B

and the plane Ax+By+C≈+D=0, are

a(Aa+Bb+ C)−a(Aa+BB+D)

∞=

Aa + Bb + C

B(Aa+Bb+C)−b(4a+BB+D)

y=

Aa+ Bb + C

Aa + BB + D

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Aa + Bb + C

When the line and plane are parallel, then

Aa+ Bb+C=0.

The equations to a perpendicular p from the point (x1,y1,1) on the plane, are

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The equation to a plane drawn through a point (~1,Y1,1) perpendicular to a given line, is

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The angle contained between the line 7, and the plane P;

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(27.) The inclination of a plane, P to the co-ordinate planes:

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(cos P,xy)2 + (cos P,xx)2 + (cos P,yz)2=1.

The mutual inclination of two planes P1, P2;

2xy + cos P1,x≈ . cos P2,≈≈

cos P1,P2 = cos P1,xy. cos Pas 19

2

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(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, Ass, Aysɔ the areas of its projections on the planes xy, xx, yx respectively,

then

If A, A, system, then

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be projected on the plane y, of some new

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If A1, A2, A3, &c. are any areas in different planes,

X, Y, Z, the sums of their projections on yz, xz, xy;

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be

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X1 = X cos x1,x + Y cos x1,y + Z cos x1,8,

Y1 = X cos y1,+ Ycos y,,y + Z cos y1,,

Z1 = X cos x1, + Y cos xy + Z cos ≈15.

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If the triangle formed by joining any three points in space 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

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

OBLIQUE CO-ORDINATES.

(29.) The equations to the line referred to oblique co

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

= x2 + y2 + x2 + 2(xy cos x,y + x ≈ cos x,z + y z cos y,*). The distance between the two points (x1,y1,1), (~29Y29%2) = (x1 − ∞2)2 + (y1 — Y2)2 + (≈, — ≈ ̧)2

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+2{(x, − x2)(y1 — y2) cos x,y + (x1 − x ̧)(≈1 — ≈2) cos x,x

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а ̧α2+b2b2+1+(a,b2+ab1)cos x,y+(a1+a2)cosx,≈+(b1+b2)cosy,≈

2

1

{ a12 + b2 + 1 + 2 (a ̧b ̧ cos x‚y + a ̧ cos x,≈ + b ̧ cos y,x)}

2

2

1

× {a ̧2 + b22 + 1 + 2(ab2 cos x‚y + a, cos x,≈ + b ̧ cos y,≈)}

2

2

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

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

THE TRANSFORMATION OF CO-ORDINATES.

(30.) Let x, y, z, be the original, 1, y1, 1, the new co-ordi-
nates, then

x=

y=

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1

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1

(x1 sin x1,xy+y1, sin y1,xy +≈, sin ≈1,xy). sin z,xy

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

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