Now the semi-axes are the greatest and least values of the semi-diameter. We have then to make

442

abc

r2 = a cos A. x2+b cos B. y2+c cos C. z2.

.....(4)

a maximum or minimum, x, y, z being connected by the relations (2) and (3).

Multiply (2) by the indeterminate multiplier 2, (4) by , adding them to (3), differentiating, and equating to zero the coefficients of each differential, we get

=

.....(5).

ux + w'y + v'z + λa cos A. x + μa: w'x + vy + u'z + λb cos B . y + μb = 0 v'x + u'y + wz + λc cos C.z + μc = 0) Multiplying these equations in order by x, y, z, and add

Substituting this value of λ in equations (5), and eliminating x, y, z from the equations combined with (2), we obtain the following quadratic for the determination of

1

2.2

:

— abcs2 (a cos B cos C+b cos C cos A+ cos A cos B),

Hence the product of the two values of 2 is

From the above investigation may be obtained the criterion which determines whether the conic be an ellipse or hyperbola. For, in the hyperbola, the two values of have opposite signs, hence the curve will be an ellipse or hyperbola according as

is negative or positive; or according as

Ua2 + Vb2 + Wc+2 U'bc + 2 V'ca + 2 W'ab

is positive or negative.

1.

EXAMPLES.

Each angular point of a triangle is joined with each of two given points; prove that the six points of intersection of the joining lines with the opposite sides of the triangle lie in a conic.

2. A conic is described, touching three given straight lines and passing through a given point; prove that the locus of its centre is a conic.

Express, in geometrical language, the position of the given point relatively to the straight lines, in order that the locus of the centre may be a circle.

Also find the locus of the given point, in order that the locus of the centre may be a rectangular hyperbola.

3. In example 1, prove that, if the conic described about the triangle, and passing through the two given points, touch the line (l, m, n), the conic passing through the six points of intersec1 tion will touch the line

(

m

4. If A, B, C, A, B, C be six points, such that the straight lines B'C', C'A', A'B' are the several polars of the points A, B, C, with respect to a given conic, prove that

The three straight lines AA', BB, CC′, intersect in a point; and that

The points of intersection of BC with B'C', CA with C'A', AB with A'B', lie in a straight line.

5. If two triangles circumscribe a conic, their angular points lie in another conic.

6. The equation of a conic circumscribing the triangle of reference, and having its semi-diameters parallel to the sides equal to r1, r,, r, respectively, is

7. A conic always touches the sides of a given triangle; prove that, if the sum of the squares on its axes be given, the locus of its centre is a circle, the centre of which is the point of intersection of the perpendiculars let fall from the angular points of the triangle on the opposite sides.

8. If be the angle between the asymptotes of the conic, represented by the general equation of the second degree, prove that

CHAPTER V.

TRIANGULAR CO-ORDINATES.

1. We shall now give a concise account of a system of co-ordinates which differs from that which has been the subject of the preceding chapters in assigning a slightly different interpretation to the co-ordinates. In the system which we are about to explain, the position of a point P is considered as determined by the ratios of the areas of the triangles PBC, PCA, PAB, to the triangle of reference ABC. If these quantities be denoted by the letters x, y, z, they will be connected by the identical relation

x+y+z=1.

2. In this method, as in that of trilinear co-ordinates, an equation of the first degree represents a straight line, and one of the second degree a conic.

Again, since : aa :: y: bẞ:: z cy, it follows that if the same straight line be represented in the two systems by the equations

la + mB + ny = 0,

l'x + m'y + n'z = 0;

.. l: l'a: m : m'b :: n: n'c.

Hence we may pass from any relation among the coefficients in the trilinear system to that in the present one, by writing la, mb, nc, for l, m, n, respectively. Similarly, in conics, we may pass from any such formula to the corresponding one, by writing

ua2, vb2, wc2, u'bc, v'ca, w'ab, for u, v, w, u', v', w'