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Similarly

d2p

x cos a―y sin a..

da2

1 2

(4)

represents a straight line through the point of intersection of two contiguous positions of the line PÅP2 and perpendicular to PP2, viz., the line PP,, and so on for further differentiations.

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200. Tangential Equation of a Curve.

DEF. The tangential equation of a curve is the condition that the line lx+my+n=0 may touch the curve. Method 1. Let F(x, y)=0 be the curve, then the tangent at x, y is

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0.

Comparing this with X+my+n=0,

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If x, y, λ be eliminated between these equations, and F(x, y)=0, or lx+my+n=0, a relation between l, m, n will result. This is the equation required.

Method 2. We may also proceed thus. Eliminate y between F(x, y)=0 and lx+my+n=0; we obtain an

equation in x, say p(x)=0. For tangency this equation must have a pair of equal roots. The condition for this will be found by eliminating x between 4(x)=0 and p′(x)=0.

In following this method, instead of eliminating y it is often better to make a homogeneous equation between F(x, y)=0 and lx+my+n=0, and then express that the resulting equation for the ratio y: x has a pair of equal roots.

Ex. Find the tangential equation of the conic

ax2+2hxy+by2+2gx+2fy+c=0.

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Al2 + Bm2 + Cn2+2Fmn+2Gnl+2Hlm=0,

where A, B, C, ... are the minors of the determinant

a, h, g

h, b, f

g, f, c

INVERSION.

201. DEF. Let O be the pole, and suppose any point P be given; then if a second point Q be taken on OP, or OP produced, such that OP. OQ= constant, k2 say, then Qis said to be the inverse of the point P with respect to a circle of radius k and centre 0.

If the point P move in any given manner, the path of Q is said to be inverse to the path of P. If (r, 0) be the polar co-ordinates of the point P, and (r', 0) those of the inverse point Q, then rr'=2. Hence, if the locus of P

be f(r, 0) = 0, that of Q will be f(2, 0)

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For example, the curves m=am cos me and cos me=a” are inverse to each other with regard to a circle of radius a.

202. Again, if (x, y) be the Cartesian co-ordinates of P. and (x', y') those of Q, then

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Hence, if the locus of P be given in Cartesians as

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Ex. The inverse of the straight line x=a with regard to a circle radius k and centre at the origin is

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a circle which touches the axis of y at the origin.

203. Tangents to Curve and Inverse inclined to Radius Vector at Supplementary Angles.

If P, P' be two contiguous points on a curve, and Q, Q' the inverse points, then, since OP. OQ=OP'. OQ', the points P, P, Q, Q are concyclic; and since the angles

OPT and OQ'T are therefore supplementary, it follows that in the limit when P' ultimately coincides with P

P

T

Fig. 28.

and Qwith Q the tangents at P and Q make supplementary angles with OPQ.

The ultimate ratio of corresponding elementary arcs,

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204. Mechanical Construction of the Inverse of a Curve. In the accompanying figure AC, CB, BQ, QA, PA, PB

is a system of freely jointed rods, of which AC= BC, and AQ=QB=BP=PA,

At P and Q sockets are placed to carry tracing pencils. A pin fixes C to the drawing board. The system is then movable about C. It is clear from elementary geometry that C, Q, P are in a straight line, and that

CP .CQ=CA2 — AQ2,

and is therefore constant.

Hence whatever curve P is made to trace out, Q will trace out its inverse, the point C being the pole of inversion.

In the figure P is represented as tracing a straight line, in which case Q will trace an arc of a circle, as shown in Art. 202.

Peaucellier has utilized this construction for the conversion of circular into rectilinear motion.

POLAR RECIPROCALS.

205. Polar Reciprocal of a Curve with regard to a given Circle.

DEF. If OY be the perpendicular from the pole upon the tangent to a given curve, and if a point Z be taken on OY or OY produced such that OY. OZ is constant (=k2 say), the locus of Z is called the polar reciprocal of the given curve with regard to a circle of radius k and centre at 0.

From the definition it is obvious that this curve is the inverse of the first positive pedal curve, and therefore its equation can at once be found.

Ex. Polar reciprocal of an ellipse with regard to its centre.

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the condition that px cos a + y sin a touches the curve is

p2=a2cos2a+b2sin2a.

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