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

1, y1, the co-ordinates of the same line in another system having the same origin, then

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

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

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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 x and y in the above formulæ.

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

x = a + x 19
y=b+ y1•

(H. A. G. 71–6; L. A. G. §. vi; G. G. A. Ch. iii; Biot, 86–96.)

(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

x = a + u

y=b+u

sin {x,y — (0 + a)}
sin x,y

sin (0+ a)

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)2 + (y − ẞ)2 + 2 (x − a) (y − ẞ). cos x, y = r2.

The equation between rectangular co-ordinates is

(x − a)2 + (y — ß)2 = r2.

(1)

If the axis of x, or y, passes through the centre, the equation becomes respectively,

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(1),

If the origin is in the circumference, (1), (2), become

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

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is

The general form when referred to rectangular co-ordinates

y2 + x2 + Ay+B+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

u2±2 (a cos 0 + b sin 0) u + a2 + b2 — r2 = 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 x2 + y2= r2 and the straight line yax+b may be determined from the equation

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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 (x,y,) in the circumference, is

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To draw a tangent from a given point (x,y) without the circle: the co-ordinates 1, y1, of the point of contact may be determined by the solution of the equations,

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(H. A. G. Ch. vi; H. C. S. 37–43; Biot, 115—23.)

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 PSPQ.

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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 a, the equation is

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may

=

The intersection of the straight line y = ax + b,

and the parabola y2=4mx; be determined by the solution of the equation

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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 (x,y,) is

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The subnormal, NG=24S=2m.

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

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The polar equation, the focus being the pole, is

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

SP.SA-SY2.

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

X

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