COR. Hence may be obtained the condition that the conic may be a parabola. For the centre of a parabola is infinitely distant, its co-ordinates must therefore satisfy the algebraical relation

This will be observed by reference to Art. 9, to be identical with the condition that the conic should touch the straight line, aa+bB+cy=0, and thus we are again led to the conclusion noticed in Art. 7, that every parabola touches the line at infinity.

9. To find the condition that the conic should touch a given straight line.

If the straight line (l, m, n) be a tangent to the conic, the values of the ratio By, obtained by eliminating a between the equation of the conic and the equation

la + mß + ny = 0,

must be equal to one another. For this purpose, it is most convenient to take the equation of the conic in the form

± (La)* ± (MB)* ± (Ny)* = 0.

or

Eliminating a, we then get

L (mB+ny) + 1 {(MB)* ± (Ny)1}2 = 0,

(Lm + Ml) ß + (Ln + Nl) y ± 21 (MNBɣ)2 = 0,

and, if the roots of this, considered as a quadratic in

It hence appears that the condition, that the line (l, m, n) should touch the conic

(La)3 ± (MB)3 + (Ny)3 = 0,

is identical with the condition that the point (l, m, n) should lie in the conic

a result analogous to that obtained in Art. 13, Chap. I.

10. To find the equations of the four circles which touch the three sides of the triangle of reference.

These may be obtained most readily by the employment of the equations for the determination of the centre, obtained in Art. 8. Thus, let it be required to find the ratios of L, M, N in order that the conic may become the inscribed circle. At the centre of this circle we have, as we know, a = B=y.

=

This gives, by the result of Art. 8,

Nb+ McLc + Na

= Ma + Lb.

To solve these equations, put each member equal to r, we then get

Adding together the last two of these equations, and subtracting the first, we get

Similar expressions being obtained for M and N, we see that

It may similarly be proved that the escribed circles, of which the centres are respectively given by

We e may remark that, at every point in the circle which touches BC externally, a is essentially negative, so that the form (-a) represents a real quantity. Similarly the appearance of (-8), (-7)* in the equations of the other two escribed circles may be accounted for*.

11. The next form of the general equation of the second degree which we propose to consider is that in which u', v', w', the respective coefficients of 28y, 2ya, 2aß, are all = 0. The equation then assumes the form

ua2 + vB2 + wy" = 0.

We observe in the first place, that if this equation represent a real conic, the coefficients of a2, B", y2, cannot be all of the same sign. Suppose the coefficient of a to be of a different sign from the other two, then writing, for convenience of future investigations, L, M, N for u, v, w respectively, our equation assumes the form

12. We have now to enquire how this conic is related to the triangle of reference.

* If these equations be rationalised, and the sines and cosines of be expressed in terms of the sides, they assume the following forms: a2 ( s − a)2 a2 + b2 (s − b)2 ß2 + c2 (s − c)2 y2 – 2bc (s – b) (s –

A B C

2' 2

c) By – 2ca (s – c) (s — a) ya – 2ab (s − a) (s – b) aß=0,

a2s2a2+b2 (s - c)2 ß2 + c2 ( s − b)2 y2 – 2bc (s – b) (s – c) By + 2cas (s − b) ya

+2abs (sc) aẞ=0,

The interpretation of this equation is, that the two straight lines drawn from B to the points in which the conic is cut by CA, form, with BC, BA, an harmonic pencil.

It may similarly be proved that the two straight lines drawn from C to the points in which the conic is cut by AB, form, with CA, CB, an harmonic pencil.

shewing that BC cuts the conic in two imaginary points. The analytical condition of harmonic section is, however, satisfied here also.

13. We may next investigate the equations of the tangents drawn through the points A, B, C.

If in the equation of the conic we put La= Ny, we get B=0, shewing that the straight line La-Ny=0 meets the conic in two coincident points, and, therefore, touches it.

Similarly

La+Ny=0, La-MB-0, La+MB=0,

are tangents to the conic.

The tangents to the conic drawn through A would be analytically represented by the equations

MB=√(−1) Nyỵ, MB=-√(−1) Ny,

which shew that these tangents are imaginary, or that the point A lies within the concavity of the conic.

14. Since the two tangents drawn through B meet the conic in points situated in the line CA, it follows that CA is the chord of contact of tangents to the conic drawn through B, or that CA is the polar of B, and B the pole of CA with respect to the conic. Similarly, C, AB, stand to one another in the relation of pole and polar.

Again, since the pole of AB is the point C, and the pole of AC is the point B, it follows that the line joining B and C is the polar of the point of intersection of Å B, AČ, i.e. that A is the pole of BC, and BC the polar of A.