Where this meets the conic, we have

L2 (mß + ny)2 + 12 (M2ß2 + N3y2) = 0,

and, making the two values of B: y equal, we get (L3m2 + M2l2) (L3n2 + N3l2) = L*m2n2,

bola.

17. To find the condition that the conic may be a para

Since every parabola satisfies the analytical condition of touching the line

18. To find the co-ordinates of the centre.

Let B, B, be the points in which the conic is cut by CA, then, if B, B, be bisected in Q, the line BQ will pass through the centre.

Now, let f,, 0, h, be the co-ordinates of B,

Now f, f, are the values of a given by the equations

which, eliminating ß, y, are equivalent to

L3c3a2 + N2 (aα — 2A)2 = 0,

This gives one straight line on which the centre lies. It may be similarly proved to lie on the straight line

Therefore the co-ordinates of the centre are given by the equations

L'a M2ß N2y

we get for the co-ordinates of the centre

Each of these becomes infinite when the conic is a parabola, as manifestly ought to be the case.

19. To find the equation of the circle with respect to which the triangle of reference is self-conjugate.

It is a distinguishing property of the circle that the line joining the centre with any other point is perpendicular to the polar of that point. Hence the line

which joins the centre with the point A, must be perpendicular to a=0. This gives (see Art. 5, p. 8)

Similarly, since the lines joining the centre with B, C are respectively perpendicular to

B=0, y=0,

we shall have

or

N'
C cos C

=

a cos Д' a cos Ab cos B

Hence the equation of the required circle is

a cos A. a2 + b cos B. B2+c cos C. y' = 0,

2

sin 24. a2 + sin 2B. B+ sin 2C. y=0.

It will be remarked that this circle will be imaginary, unless one of the quantities sin 2A, sin 2B, sin 2C be negative, that is, unless one of the angles 2A, 2B, 2C be greater than two right angles, or unless the triangle of reference be obtuse-angled.

COR. By referring to the expressions for the co-ordinates of the centre of the conic, given in Art. 17, we see that at the centre of the circle we have

Or, the centre of the circle, with respect to which the triangle of reference is self-conjugate, coincides with the intersection of the perpendiculars drawn from the angular points to the opposite sides. This is otherwise evident from geometrical considerations.

20. To find the equation of the conic which touches two sides of the triangle of reference in the points where they meet the third.

Let AB, AC be the two sides which the required conic touches in the points B, C. We then require that the constants in the equation

La2 + MB2 + Ny2+ 2λ By + lμ ya+2v aß = 0

should be so related to one another, that when ẞ=0 we have the two values of a= 0, and also when y = 0 the two values of a may each = 0.

Hence the two equations

La2 + Ny2+2μ ya= 0,

La2+MB2+2v aß =0,

must both be identically satisfied when a = 0, and by no other value. This requires that

N=0, μ=0, M=0, v=0.

Hence the equation reduces to

This equation, it will be observed, involves only one arbitrary constant, as ought to be the case, since when a tangent and its point of contact are given, the conic is thus subjected

to two conditions, and, therefore, when two tangents and their points of contact are given, to four.

21. If any straight line whatever be drawn through A, and meet the conic in P, Q, and be represented by the equation

B = n2y,

then BP, BQ will be represented by the equations

from the form of which it is apparent that BA, BP, BC, BQ form an harmonic pencil. Or any chord of a conic is divided harmonically by the conic itself, any point on the chord, and the polar of the point with respect to the conic.

22. We may observe, that the two straight lines represented by the equations

ka=wß, ka=1y,

intersect on this conic whatever be the value of w. Hence any point on the conic may be expressed by giving the value of the ratio

If a be the value of this ratio at any point, that point may be denoted by the letter w*. The line joining the two points w, w' may be called the line ww'.

23. To find the equation of the line ww'.
Let the required equation be

ka+mB+ny = 0,

we have then to determine m and n.

* This mode of expression is given by Salmon in his Conic Sections.