hence the polar of any point on E is a tangent to L. That is, if a point move along E, its polar will envelope L. In other words, I may be generated from E, as E was from L. In this consists the reciprocity of the curves.

The process of generating E from L, or L from E, is called reciprocating L or E.

4. If the curve L be cut by any straight line whatever, the polars of the several points of intersection will be the several tangents to E, drawn through the pole of the cutting line. And conversely, the several tangents drawn to L from any point will have for their poles the several points in which E is intersected by the polar of that point.

If any two curves be reciprocated, the polar of any point common to both will be a common tangent to the reciprocal curves, and the pole of any tangent common to both will be a point of intersection of the reciprocal curves. Hence any two curves will have as many points of intersection as their reciprocals have common tangents, and as many common tangents as their reciprocals have points of intersection.

If the curves touch one another, then two of their points of intersection coincide; and consequently the two corresponding tangents to the reciprocal curves will coincide, and therefore the reciprocal curves will also touch one another.

5. From what has been said above, it will be seen that the total number of tangents, real or imaginary, which can be drawn to E or L from any point (not on the curve itself) is equal to the total number of points, real or imaginary, in which L or E is cut by any straight line, not a tangent to it.

A curve, to which n tangents can be drawn through the same point, is said to be of the nth class, and we may therefore express the above proposition by saying that the degree of a curve is the same as the class of its reciprocal, and the class of a curve the same as the degree of its reciprocal.

6. The number of tangents, real or imaginary, which can be drawn to a conic from a given point is known to be two. Hence, the reciprocal of a conic is intersected by a given straight line in two points, real or imaginary, and is therefore of the second degree, that is, it is itself a conic.

7. This proposition may also be proved analytically as follows.

DEF. The conic with respect to which the poles and polars are taken is called the auxiliary conic.

We have seen (Art. 15, Chap. II.) that any two conics may be expressed by equations involving the squares of the variables only. Let then the auxiliary conic be denoted by Lx2 + MB2 + Ny2 = 0.......................(1),

and the conic to be reciprocated by

.........

la2 + mß2 + ny2 = 0.........(2).

If (f, g, h) be any point on the required curve, its polar with respect to (1) will be given by the equation

Lfa+MgB+ Nhy = 0.

In order that this may touch (2) we must have (see Art. 16, Chap. II.)

L2

N2

ī

M2 -g2+

h2 = 0...... (3).

m

n

(3), regarding f, g, h as current co-ordinates, is therefore the reciprocal of (2) with respect to (1).

COR. It hence appears that the three points which form a conjugate triad for two given conics, will also form a conjugate triad for the reciprocal of one with respect to the other.

8. To find the polar reciprocal of the conic
ux2 + vß2 + wy3 + Qu' By + 2v'ya + 2w'aß=0

[The conic, a2+B+y=0, is imaginary, but the analytical process of finding the pole of a given straight line, or the polar of a given point, may be equally well performed, whether the auxiliary conic be imaginary or real, provided its coefficients be real.]

Let f, g, h be any point on the required locus, its pole with respect to the auxiliary conic is

fa+gB+hy = 0,

and in order that this may touch the given conic, we must have (Art. 4, Chap. IV.)

0, f, g, h

f, u, w', v'

g, w', v, u'

h, v', u', w

2

or, (vw — u'2) ƒ2 + (wu — v22) g2 + (uv — w12) h2

+2 (v'w' — uu') gh + 2 (w'u' — vv') hf + 2 (u'v' — ww') fg = 0, which, adopting the notation of Chap. IV., may be written Uƒ2 + Vg2 + Wh2 + 2 U'gh + 2 V'hf +2 W'fg = 0.

This is therefore the required equation.

It may be proved, in a similar manner, that if

$(α, B, y) = 0, & (α, B, y) = 0

be the equations of any two conics, the equation of the reciprocal of the first with respect to the second is

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, without, or upon, the original conic.

10. We have now sufficient materials for transforming 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

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', Q', 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