as having four axes of similitude. (See Art. 117, of which this theorem is an extension).

If from any point on the tangent at the point of contact of two conics which touch, a tangent be drawn to each, the line joining their points of contact will pass through the intersection of common tangents to the conics.

If, on a common chord of two conics, any two points be taken, and from these tangents be drawn to the conics, the diagonals of the quadrilateral so formed will pass through one or other of the intersections of common tangents to the conics.

If A and B be two conics having each double contact with S, the intersections of the tangents at their points of contact with S, and the intersections of tangents common to A and B, lie in one right line, which they divide harmonically.

If A, B, C, be three conics, having each double contact with S, and if A and B both touch C, the line joining the points of contact will pass through an intersection of common tangents of A and B.

307. We have hitherto supposed the auxiliary conic U to be any conic whatever. It is most common, however, to suppose this conic a circle; and hereafter, when we speak of polar curves, we intend the reader to understand polars with regard to a circle, unless we expressly state otherwise.

We know (Art. 88) that the polar of any point with regard to a circle is perpendicular to the line joining this point to the centre, and that the distances of the point and its polar are, when multiplied together, equal to the square of the radius; hence the relation between polar curves with regard to a circle is often stated as follows: Being given any point O, if from it we let fall a perpendicular OT on any tangent to a curve S, and produce it until the rectangle OT. Op is equal to a constant k2, then the locus of the point p is a curve s, which is called the polar reciprocal of S. For this is evidently

T

R

T'

S

P'

S

equivalent to saying that p is the pole of PT, with regard to a circle whose centre is 0 and radius k. We see, therefore (Art. 301), that the tangent pt will correspond to the point of contact P, that is to say, that OP will be perpendicular to pt, and that OP. Ot=k2.

It is easy to show that a change in the magnitude of k will affect only the size and not the shape of s, which is all that in most cases concerns us. In this manner of considering polars, all mention of the circle may be suppressed, and s may be called the reciprocal of S with regard to the point 0. We shall call this point the origin.

The advantage of using the circle for our auxiliary conic chiefly arises from the two following theorems, which are at once deduced from what has been said, and which enable us to transform, by this method, not only theorems of position, but also theorems involving the magnitude of lines and angles:

The distance of any point Pfrom the origin is the reciprocal of the distance from the origin of the corresponding line pt.

The angle TQT between any two lines TQ, T'Q, is equal to the angle pop' subtended at the origin by the corresponding points P, p'; for Op is perpendicular to TQ, and Op' to T'Q.

We shall give some examples of the application of these principles when we have first investigated the following problem:

308. To find the polar reciprocal of one circle with regard to another. That is to say, to find the locus of the pole p with regard to the circle (0) of any tangent PT to the circle (C). Let MN be the polar of the point C

M'

distance of p from O is to its distance from MN in the constant ratio OC:CP; its locus is therefore a conic, of which O is a focus,

MN the corresponding directrix, and whose eccentricity is OC

divided by CP. Hence the eccentricity is greater, less than, or = 1, according as O is without, within, or on the circle C.

Hence the polar reciprocal of a circle is a conic section, of which the origin is the focus, the line corresponding to the centre is the directrix, and which is an ellipse, hyperbola, or parabola, according as the origin is within, without, or on the circle.

309. We shall now deduce some properties concerning angles, by the help of the last theorem given in Art. 307.

Any two tangents to a circle make equal angles with their chord of contact.

The line drawn from the focus to the intersection of two tangents bisects the angle subtended at the focus by their chord of contact. (Art. 191).

For the angle between one tangent PQ (see fig., p. 270) and the chord of contact PP' is equal to the angle subtended at the focus by the corresponding points p, q; and similarly, the angle QP'P is equal to the angle subtended by p', q; therefore, since QPP'=QP'P, pOq=p'Oq.

Any tangent to a circle is perpendicular to the line joining its point of contact to the centre.

Any point on a conic, and the point where its tangent meets the directrix, subtend a right angle at the focus.

This follows as before, recollecting that the directrix of the conic answers to the centre of the circle.

Any line is perpendicular to the line joining its pole to the centre of the circle.

The line joining any point to the centre of a circle makes equal angles with the tangents through that point.

The locus of the intersection of tangents to a circle, which cut at a given angle, is a concentric circle.

The envelope of the chord of contact of tangents which cut at a given angle is a concentric circle.

If from a fixed point tangents be drawn to a series of concentric circles, the locus of the points of contact will be a circle passing through the fixed point, and through the common centre.

Any point and the intersection of its polar with the directrix subtend a right angle at the focus.

If the point where any line meets the directrix be joined to the focus, the joining line will bisect the angle between the focal radii to the points where the given line meets the curve.

The envelope of a chord of a conic, which subtends a given angle at the focus, is a conic having the same focus and the same directrix.

The locus of the intersection of tangents, whose chord subtends a given angle at the focus, is a conic having the same focus and directrix.

If a fixed line intersect a series of conics having the same focus and same directrix, the envelope of the tangents to the conics, at the points where this line meets them, will be a conic having the same focus, and touching both the fixed line and the common directrix.

In the latter theorem, if the fixed line be at infinity, we find the envelope of the asymptotes of a series of hyperbolas having the same focus and same directrix, to be a parabola having the same focus and touching the common directrix.

If two chords at right angles to each other be drawn through any point on a circle, the line joining their extremities passes through the centre.

The locus of the intersection of tangents to a parabola which cut at right angles is the directrix.

We say a parabola, for, the point through which the chords of the circle are drawn being taken for origin, the polar of the circle is a parabola (Art. 308).

The envelope of a chord of a circle which subtends a given angle at a given point on the curve is a concentric circle. 2.41

Given base and vertical angle of a triangle, the locus of vertex is a circle passing through the extremities of the base.

The locus of the intersection of tangents to a parabola, which cut at a given angle, is a conic having the same focus and the same directrix.

Given in position two sides of a triangle, and the angle subtended by the base at a given point, the envelope of the base is a conic, of which that point is a focus, and to which the two given sides will be tangents.

The envelope of any chord of a conic which subtends a right angle at any fixed point is a conic, of which that point is a focus.

"If from any point on the circumference of a circle perpendiculars be let fall on the sides of any inscribed triangle, their three feet will lie in one right line" (Art. 125).

If we take the fixed point for origin, to the triangle inscribed in a circle will correspond a triangle circumscribed about a parabola; again, to the foot of the perpendicular on any line corresponds a line through the corresponding point perpendicular to the radius vector from the origin. Hence, "If we join the focus to each vertex of a triangle circumscribed about a parabola, and erect perpendiculars at the vertices to the joining lines, those perpendiculars will pass through the same point." If, therefore, a circle be described, having for diameter the radius vector from the focus to this point, it will pass through the vertices of the circumscribed triangle. Hence, Given three tangents to a parabola, the locus of the focus is the circumscribing circle (p. 196).

The locus of the foot of the perpendicular (or of a line making a constant angle with the tangent), from the focus

If from any point a radius vector be drawn to a circle, the envelope of a perpendicular to it at its extremity (or of a NN

of an ellipse or hyperbola on the tangent line making a constant angle with it) is a is a circle. conic having the fixed point for its focus.

310. Having sufficiently exemplified in the last Article the method of transforming theorems involving angles, we proceed to show that theorems involving the magnitude of lines passing through the origin are easily transformed by the help of the first theorem in Art. 307. For example, the sum (or, in some cases, the difference, if the origin be without the circle) of the perpendiculars let fall from the origin on any pair of parallel tangents to a circle is constant, and equal to the diameter of the circle.

Now, to two parallel lines correspond two points on a line passing through the origin. Hence, "the sum of the reciprocals of the segments of any focal chord of an ellipse is constant."

We know (p. 179) that this sum is four times the reciprocal of the parameter of the ellipse, and since we learn from the present example that it only depends on the diameter, and not on the position of the reciprocal circle, we infer that the reciprocals of equal circles, with regard to any origin, have the same parameter.

The rectangle under the segments of any chord of a circle through the origin is constant.

The rectangle under the perpendiculars let fall from the focus on two parallel tangents is constant.

Hence, given the tangent from the origin to a circle, we are given the conjugate axis of the reciprocal hyperbola.

Again, the theorem that the sum of the focal distances of any point on an ellipse is constant, may be expressed thus:

The sum of the distances from the focus of the points of contact of parallel tangents is constant.

The sum of the reciprocals of perpendiculars let fall from any point within a circle on two tangents whose chord of contact passes through the point, is constant.

311. If we are given any homogeneous equation connecting the perpendiculars PA, PB, &c. let fall from a variable point P on fixed lines, we can transform it so as to obtain a relation connecting the perpendiculars ap, bp', &c. let fall from the fixed points a, b, &c. which correspond to the fixed lines, on the variable line which corresponds to P. For we have only to divide the equation by a power of OP, the distance of P from the origin, and then, by Art. 101, substitute for each term