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(6.) Let x, y, be the co-ordinates of

any

line, X1, , the co-ordinates of the same line in another system having the same origin, then

1

(x, sin x 7,9 +Y..sin yuy),

sin x,y

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x = N1

If the axės in the original system are rectangular, and in the new system, oblique, the above formulæ become

COS X 190 + Y, cos Y198,

y=x, sin X1 + y, sin Y1X. If the original axes are oblique, and the new ones rectangular, then

1

(x, sin Q1,4 + y, cos X1,Y),

sin x,y

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sin ,

If both systems are rectangular, then

X=N, COS X 190 - N 1980

y=x, sin x 1,60 + y, cos x 7,8. If it be required to change the origin, as well as the direction of the co-ordinates, then a, b, the co-ordinates of the new origin, must be respectively added to the values of w and y in the above formulæ.

If the origin alone be changed, the direction of the coordinates remaining the same, then

x= a + x12

y=b+yi (H. A. G. 71-6; L. A. G. . vi; G. G. A. Ch. iii; Biot, 86–96.)

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(7) Transformation of polar co-ordinates.

Let a, b, be the co-ordinates of the new pole, u, the radius vector, and a the angle which the axis to which u is referred makes with the axis of x, then

sin {x,y— (@+a)} x=a+u

a sin (O + a) y=b+u

sin X,Y

If the axes are rectangular, these formulæ become

x=a + u cos (0 + a),
y=b+ u sin (0 + a).

(H. A. G. 77-80; Biot, 100.)

THE CIRCLE.

(8.) Let a, ß, be the co-ordinates of the centre, r the radius;
then the equation to the circle is

(x – a)? + (y-2)* + 2(x - a)(y-3).cos ,y=p?.
The equation between rectangular co-ordinates is
(x - a)? + (y-B)=pl.

(1)
If the axis of X, or y, passes through the centre, the equation
becomes respectively,
X? + (y-3)=pa,

(2) or (x - a)? + = pl. If the origin is in the circumference, (1), (2), become respectively,

x2 + yo - 2aX 23y=0); (3)

y=2rx - x,

do =2g-go. If the origin is at the centre, then 2? + yo =r?.

(5) of these forms, (4) and (5) occur most frequently.

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y?

a +1% +

The general form when referred to rectangular co-ordinates is

+ x2 + 4y + Bx +C=0. (9.) The polar equation. Let a, b, be the co-ordinates of the pole, u the radius vector, and the angle u, x=0; the polar equation is

u° +2 (a cos @ + b sinu + a + b - p=0; + or acting as the pole is situated within or without the circle. (H. A. G. Ch.v; G. G. A. Ch. x; Biot 105-14.) (10.) The intersection of the circle xé + yo=yo and the straight line y=ax + b may be determined from the equation

26 6 - ape? ,

-0,

a +1 which having only two roots, it follows that a straight line cannot cut a circle in more than two points.

The equation to a tangent at a given point (x1,9.) in the circumference, is

Ny + 91 cos x,y
Y - Y1

(w — X1),

Yit in COS X,Y if the axes are rectangular,

y-Y1= (x x1), or

Yi

DX, +yy;=r? To draw a tangent from a given point (x,y) without the circle: the co-ordinates X1, , of the point of contact may be determined by the solution of the equations,

X2X1 + y2y.= gel

a + y'=p? (H. A. G. Ch. vi; H. C. S. 37-43; Biot, 115–23.)

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CONIC SECTIONS.

THE PARABOLA.

(11.) The parabola referred to its axis.

Let S be the focus, P any point in the curve, and PQ perpendicular to EQ the directrix, then PS=PQ.

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Let PT be a tangent at the point P, and PG the normal.

If the origin is at the vertex, and the axis coincides with the axis of x, the equation is

yo = 4ma,

or PN=4AS. AN.
ZQPT= LSPT.
SP= AN + AS=ST=SG.
The intersection of the straight line y = ax +b,

and the parabola yo = 4mx ; may be determined by the solution of the equation

4m 4 mb

y +

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0:

a

a

which has only two roots; hence a straight line cannot cut a parabola in more than two points.

The equation to a tangent at the point (109.) is

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The subnormal, NG=2 AS=2m.

To draw a tangent to the parabola from a given point (a,b) without it: 81,9, the co-ordinates of the point of contact may be determined from the equations

2m
b= (a + x,),

y ? = 4mx1. (H.C. S. 46–62; Hust. Prop. 3—7.) (12.) The parabola referred to the focus.

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The polar equation, the focus being the pole, is

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SP

1 + cos e (sin 0) 1

1 2 +

(See Fig. p. 160.)
SP

SL
SP.Sp=AS. Pp.

The tangent at any point, and the perpendicular on it from the focus intersect the axis AY in the same point.

SP.SA= SY

Let Z be the intersection of the tangent with the directrix, SZ is perpendicular to SP. (H.C. S. 63–71; Hust. Prop. 8.)

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