Page images
PDF
EPUB

The polar of this with respect to

La® + MB2 + Nye = 0, to which form every conic may, by suitable choice of the triangle of reference, be reduced, is represented by the equation (m,n, m,n) La + (n la nyl) MB+(l/m,-im) Ny=0...(3).

But the poles of (1) and (2) with respect to the same conic are given by

La _ MB _Ny
T

ni
La _ MB _ Ny
my

п, Both these points lie on the line (3). Hence the proposition is proved.

Ny,

ma

[ocr errors]

3. If a point move in any manner whatever, its polar will move in a manner dependent upon the motion of the point, and the curve which the polar always touches (its envelope, as it is called) will have certain definite relations to the path traced out by the point. The locus of the moving point and the envelope of its polar, are called the polar reciprocals of one another. The use of the word reciprocal arises from the fact, which we proceed to demonstrate, that the locus of the point may be generated from the envelope of its polar, in the same manner as the latter curve was generated from the former. For shortness' sake we shall denote the two curves by the letters L and E.

Let P, P' be any two points on L, the pole Q of the chord PP' will be the point of intersection of the corresponding tangents to E (that is, of the two tangents to E which are the poles of P, P' with respect to the conic). Now let P' move along L up to P, then PP' ultimately becomes the tangent to L at P; moreover the polars of P and P' approach indefinitely near to coincidence, and their point of intersection Q will ultimately be a point on E. But is the pole of PP,

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, L 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. 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.

Hence any

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 tan

gent 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. 11.) that any two conics may be expressed by equations involving the squares of the variables only. Let then the auxiliary conic be denoted by

Li + MB+ Ny = 0.........(1), and the conic to be reciprocated by

le + + nye = 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.)

L? M N2
ī
fP+

h = 0...... (3).

- go + (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.

m

n

8. To find the polar reciprocal of the conic

uz+uß? + wory' + 2u'By + 20'ya + 2w'aß= 0) with respect to

a? + B2 + y = 0.

[ocr errors]

[The conic, a2 +82 + 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+ hry=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', u'
g, w', v,

u

h, u', u', w or, (vw u'?) f? + (wu voo2) g* + (uv - w'9) h* +2 (u'w' uu') gh+2 (w'u' vo') hf +2 (u'v' ww') fg =0, which, adopting the notation of Chap. IV., may be written

Ufo + Vg* + Who + 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

$(A, B, y) = 0, y (a, b, y) = 0 be the equations of any two conics, the equation of the reciprocal of the first with respect to the second is

df dý
0.

= 0.
d' dy
df d'o d'd d'o
da' dal' dadß' da dy
dry do dd d' ¢
' dB da ?

dß dy
df d'd d' ¢
dy' dy da' dy dß' dye

df

[ocr errors]

da

[ocr errors]
[ocr errors]

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

« PreviousContinue »