reference, that each side of the triangle is the polar, with respect to the conic, of the opposite angular point*.

This is expressed by saying that the triangle is self-conjugate with respect to the conic; or that the three angular points of the triangle form a conjugate triad.

The geometrical properties of the conic having been thus established, we shall, in future investigations, write for the sake of symmetry of form, L' instead of L, so that the equation of the conic will be written

L2a2 + M2ß2 + N2y2 = 0.

It must here be borne in mind that one of the three quantities L, M, N is essentially imaginary.

15. Any two conic sections represented by such equations as

L'a2+M2ß2+Ny2 = 0,

L'2a2+M22ß2+N'2y2=0,

have important relations to one another, which we proceed to consider.

They will of course intersect in four points, which may be real or imaginary. We will first suppose them real, and represent them by the letters P, Q, R, S.

Now the locus of the equation

(L3M" — L'2M2) ßa2 + (L3N'2 — L'2N2) y2 = 0

passes through all the points P, Q, R, S; and, since it may be resolved into linear factors, represents two straight lines. Suppose them to be PQ and RS. The intersection of these two straight lines is given by the equations

*If the coefficients of p2 and y2 be equal, and the triangle of reference be right-angled at A, the form of the equation shews that A will be a focus of the conic, and BC the corresponding directrix.

(L3M'2 — L'M2)3 B = (L'N2 — L2N'2)§ y,

(L3M" — L'3M2)* ß = — (L'3N2 — L3N'2) Y,

which evidently give ß=0, y=0.

Hence PQ, RS intersect in A.

Similarly, PR, QS intersect in B, and PS, QR intersect in C. Hence, the angular points of the triangle of reference coincide with the intersections of the line joining each pair of points of intersection of the conics with the line joining the other pair. Hence also, if any number of conic sections be described about the same quadrangle*, and the diagonals of that quadrangle intersect in A, while the sides produced intersect in B and C, then A, B, C form, with respect to each of the circumscribing conics, a conjugate triad. The points A, B, C may themselves be called vertices of the quadrangle, or of the system of circumscribing conics.

any

It will be seen, from the preceding investigation, that two conics which intersect in four real points can be reduced, by a proper choice of the triangle of reference, to the form

L2a2 + M2ß2 + N3y2 = 0.

The same reduction may also be effected in every case with the reservation that if two of the points of intersection of the conics be real and two imaginary, then two of the angular points of the triangle of reference (or vertices) will be imaginary and the remaining one real. If all the points. of intersection be imaginary, the vertices of the conics will be all real. This we shall prove hereafter.

16. To find the condition that a given straight line may

touch the conic.

Let the equation of the straight line be

la+mẞ+ny = 0.

* I employ the term quadrangle in preference to quadrilateral, considering a quadrangle as a figure primarily determined by four points, a quadrilateral by four indefinite straight lines.

Where this meets the conic, we have

L2 (mB + ny)? + l2 (M2ß2 + N2y3) = 0,

and, making the two values of ß: y equal, we get (L3m2 + M2l2) (L3n2 + N2l2) = L*m3n2,

Since every parabola satisfies the analytical condition of touching the line

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

Let B1, 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 fa, fi are the values of a given by the equations

which, eliminating B, y, are equivalent to

L3c2a2 + N2 (aa — 2▲)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

La MB Ny

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

or

Hence the equation of the required circle is

a cos A. a2 + b cos B. B2+c cos C. y2 = 0,

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

It will be remarked that this circle will be imaginary, unless one of the quantities sin 2A, sin 2B, sin 2 C 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