The condition of perpendicularity,

2ll'a2+2mm'b* + 2nn'c2 - (mn' + m'n) (b2 + c2 — a2)

-(nl' + n'l) (c2 + a2 — b2)

(Im' + I'm) (a2 + b2 — c2) = 0,

or {(lm) (l' - n') + (l — n) (l' —

m')} a2

+ {(m − n) (m' — l') + (m − 1) (m' — n')} b2

+ {(n − 1) (n' — m') + (n − m) (n' — l'′) } c2 = 0.

The perpendicular distance from the point (x, y, z) to the line la+my+nz = 0, is

(lx+my+nz) 2A

{(l — m) (1 − n) a2 + (m − n) (m − 1) b2 + (n − 1) (n − m) c2}1 °

The line at infinity will be represented by x + y + z = 0. 3. Again, in conics we have the following formulæ :

The conic will be a parabola, if

or if

or

A rectangular hyperbola, if

ua2 + vb2 + wc2 — u′ (b3 + c2 — a2) — v′ (c2 + a2 — b2)

The radical axis of two circles

ux2 + vy2+wz2 + 2u'yz + 2v′zx + 2w'xy = 0,

px2 + qy2 + rz2 + 2p'yz + 2q'zx + 2r'xy = 0,

is represented by the equation

- 2 (s—b) (s—c) yz — 2 (s — c) (s — a) zx − 2 (s − a) (s — b) xy = 0.

-

The square on the tangent from the point x, y, z to the circle ux+...+2u'yz+... 0 is

a2 + b2+c2

=

(ux2+vy2+wz2+2u'yz+2v′zx+2w'xy).

Other formulæ may be adapted in a similar manner.

CHAPTER VI.

RECIPROCAL POLARS.

1. THE theory of Reciprocal Polars, which will be treated of in this chapter, discusses the relations which exist between systems of points and straight lines which are the poles and polars of each other with regard to any conic; and shews how from the properties of a curve, regarded as the locus of a moving point, may be deduced those of another curve which is always touched by the polar of this moving point with regard to a fixed conic. The theory is especially valuable when the conic, with respect to which the poles and polars are taken, is a circle.

2. The polar of the point of intersection of two given straight lines is the straight line which joins the poles of those straight lines. This will readily be seen to follow geometrically from the definitions of a pole and polar; or it may be analytically proved thus.

Let the two straight lines be represented by the equations

l1a+m‚ß + n1y=0........
l2a+m2ẞ + n2y=0............

At their point of intersection, we have

(1),

(2).

The polar of this with respect to

La2 + MB2 + Ny2 = 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 ̧l2 − n ̧1 ̧) Mß + (1 ̧m2 — l ̧m ̧) Ny = 0...(3).

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

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

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 polars of P, P' with respect to the conic). Now let P' move along L up to P, then PP' ultimately becomes the tangent to Lat 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 Q is the pole of PP',

In

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. other words, I may be generated from E, as E was from L. In this consists the reciprocity of the curves.

L

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 of 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.