9. Since the tangents at the extremity of any diameter of a conic are parallel to one another, it follows that the polar of the centre is at an infinite distance, and conversely, that the line at infinity reciprocates into the centre of the auxiliary conic. Hence it follows that parallel lines reciprocate into points lying on a straight line passing through the centre of the auxiliary conic; and that the asymptotes of any curve, being the tangents drawn to it at the points. where it meets the line at infinity, reciprocate into the points of contact of the tangents drawn to the reciprocal curve from the centre of the auxiliary conic.

Since the asymptotes of an hyperbola are real, while those of an ellipse are imaginary, it follows that the tangents, drawn from the centre of the auxiliary conic (supposed real) to the reciprocal curve, will be real or imaginary, according as the original curve is an hyperbola or an ellipse. If it be a parabola, the reciprocal curve will pass through the centre of the conic, which is in accordance with what has already been stated, that every parabola touches the line at infinity. Conversely, if one conic be reciprocated with respect to another, the reciprocal curve will be an ellipse, parabola, or hyperbola, according as the centre of the auxiliary conic lies within, upon, or without, the original conic.

10. We have now sufficient materials for transforming any descriptive proposition, that is any proposition relating to the position of lines and points, without reference to considerations of magnitude, into another. Before proceeding further, we will give a few examples of this process.

We will first take the following proposition. "If two of the angular points of a triangle move each along a fixed straight line, and each side pass through a fixed point, the three points lying in the same straight line, the third angular point will move along a straight line, passing through the intersection of the straight lines along which the other angular points move."

The reciprocals of the three sides of the given triangle will be three points, which may be considered as the angles of a triangle, which may be called the reciprocal triangle. Those

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of the angular points of the first triangle will be the sides of the reciprocal. Those of the fixed straight lines, along which two of the angular points of the first triangle move, will be fixed points through which two of the sides of the reciprocal triangle pass. Those of the three points, lying in the same straight line, through which the sides of the given triangle always pass, will be three straight lines, intersecting in a point, along which the angular points of the reciprocal triangle always move. Hence the data of the reciprocal proposition will be "Two of the sides of a triangle pass each through a fixed point, and each angular point moves along a fixed straight line, the three straight lines passing through the same point." In the given theorem, the thing to be proved relates to the motion of the third angular point. To this will correspond the third side of the reciprocal triangle. To the straight line, passing through the intersection of the two given straight lines, along which the third angular point may be shewn to move, corresponds a point lying in the same straight line with the two given points, and through this the third side will always pass. Hence, under the circumstances stated above as data of the reciprocal theorem, "the third side will pass through a fixed point lying in the straight line joining the two fixed points, through which the first sides pass* * ""

* The given theorem may be expressed, by the aid of letters, as follows: Let PQR be the given triangle, and let its angular point Q move along a fixed straight line OX, its angular point R along a fixed straight line OY. Also, let the straight line QR always pass through a fixed point F, RP through a fixed point G, PQ through a fixed point H, the three points F, G, H lying in the same straight line. Then the given theorem tells us that the point P will always move along a fixed straight line, passing through 0.

Now let the whole figure be reciprocated with respect to any conic section. Let the line which is the polar of any point be denoted by accenting the same single letter by which the point is denoted in the original figure; the polar of P, for example, being denoted by P'. Then the point of intersection of the lines P', 'will be denoted by the two letters P' Q', and this will be the pole of the line PQ. We have then a triangle of which the sides are P', Q', R', the side Q'always passing through a fixed point O'X', the side R' through a fixed point O'Y. Also the angular point Q'R' always moves along a fixed straight line F, the point R'P' along a fixed straight line G', the point P'Q along a fixed straight line H', the three straight lines F', G', H' passing through the same point. Then the reciprocal theorem is that the side P' will always pass through a fixed point lying in the line Q'.

The student will find the above mode of transformation, in which a

Again, turn to Example 4, on page 57, and let us investigate the reciprocal theorem. The three conics touching respectively each pair of the sides of a triangle at the angular points where they meet the third side, will reciprocate into "three conics passing respectively through each pair of the angular points of a triangle, and touching the lines joining them with the third angular point," that is, the sides of the triangle themselves. This condition, therefore, reciprocates into itself. The condition "all intersecting in a point" reciprocates into "all touching a straight line." Hence the data are, "Three conics are drawn, touching respectively each pair of the angular points of the sides of a triangle at the points where they meet the third side, and all touching a straight line."

In the matter to be proved, we may first enquire what are the reciprocals of "the sides of the triangle which intersect" (that is, which do not touch) "their respective conics." These will be "the angular points of the triangle not lying on their respective conics." The three tangents at their common point will reciprocate into "the three points of contact of their common tangents." And the meeting of the tangents with the sides will reciprocate into the lines joining the points of contact with the angular points. Hence the first thing to be proved is, "That the three straight lines joining the points of contact of the common tangent with the angular points of the triangle not lying on the respective conics all pass through a point."

Again, "the other common tangents to each pair of conics" reciprocate into "the other points of intersection of each pair of conics," and "the sides of the triangle which touch the several pairs of conics" into the angular points of the triangle "common to the several pairs of conics." Hence the latter part of the theorem will run: "And that the same three straight lines respectively join the other point of intersection of each pair of conics with the angular point of the triangle common to each pair."

straight line is denoted by a single letter, and a point by the pair of letters representing any two straight lines which intersect in it, a useful mode of familiarizing himself with the method of reciprocal polars.

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11. After a little practice, the process of reciprocating a given theorem will be found to consist simply in writing "straight line" for "point," "join" for "intersect," "locus" for "envelope," &c., and vice versa. The word "conic"

will of course remain unaltered.

12. Brianchon's Theorem.

By reciprocating Pascal's Theorem (given in Art. 12, Chap. III.), we obtain Brianchon's Theorem, which asserts that

"If a hexagon be described about a conic section, the three diagonals will intersect in a point*."

* It may be well to append an independent proof of this important theorem. Take three sides of the hexagon as lines of reference, and let the equations of the other three be

a+m1ß+n ̧y=0, l2a+ß+n,y=0, lзa+m3ß+y=0.

Let the equation of the conic be

The line passing through the intersections of B=0 with (l ̧, m2, 1) and of y=0 with (12, 1, n) is represented by the equation

+y=0;

The student will find it useful to transform, by the method of reciprocal polars, the special cases of Pascal's Theorem, given in Art. 13, Chap. III.; and to obtain a geometrical construction by which when five tangents to a conic are given, their points of contact may be found.

13. The anharmonic ratio of the pencil formed by four intersecting straight lines is the same as that of the range formed by their poles. This may be proved as follows.

Let OP, OQ, OR, OS be the four straight lines, P', ', R', S' their poles, which will lie in a straight line, the polar

OA

PO'R'

Fig. 18.

of O; let P, Q, R, S be the points in which the pencil is cut by the transversal P'Q'R'S.

Let this transversal cut the conic in K1, K. Bisect