the straight line

From the form of these equations it will be seen that

passes through D, E, F. Hence these three points are in a straight line.

COR. The converse proposition to that above enunciated may be demonstrated by similar reasoning.

The point P, and the line DEF, may be called harmonics of one another with respect to the triangle ABC.

By combining the proposition last proved with that proved in Art. (22), we shall obtain a demonstration of the statements made in Art. 6; that the points in which the external bisectors of each angle of a triangle respectively intersect the sides opposite to them, lie in the same straight line; and that the points in which the external bisector of any one angle and the internal bisectors of the other two angles, intersect the sides respectively opposite to them, lie in the same straight line.

These straight lines will be respectively represented by the equations,

a+B+y=0, B+y-a=0,

y+a-B=0, a+ B-y=0.

ON INVOLUTION.

26. DEFS. Let O be a point in a given straight line, and let

P, P, Q, Q, R, R'......

be a series of points on that line so taken that

Then these points are said to form a system in involution.

If K be a point such that OK = k2, K is called a focus of the system.

If 2 be positive, there will evidently be two such foci, one on each side of O, if negative (and k therefore imaginary) there will be no real foci.

The point O is called the centre of the system.

Two points, such as P, P', are said to be conjugate to one another.

It is evident that each focus is conjugate to itself, and that the conjugate of the centre is at an infinite distance, and that a point and its conjugate will be on the same, or different sides of the centre, according as the foci are real or imaginary.

The system will be determined when two foci, or a centre and focus, are given. It will also be determined if two pair of conjugate points be given; as may be seen as follows.

Let p, p, q, q' be the respective distances of the four points from any arbitrary point on the line, x the distance of the centre from the same point.

27. PROP. The anharmonic ratio of four points is equal to that of their four conjugates.

For, if OP=p, OQ=q, OR=r, OS=s,

COR. It is evident that [PQRP'] = [P'Q'R'P].

28. PROP. Any two conjugate points form, with the two foci, an harmonic range.

or the four points in question form a harmonic range.

Conversely, if there be a system of pairs of points in a straight line, such that each pair forms, with two given points, an harmonic range, the aggregate of the pairs of points will form a system in involution, of which the two given points are the foci.

29. A system of straight lines, intersecting in a point, may be treated in the same manner as a system of points lying in a straight line, the sine of the angle between any two lines taking the place of the mutual distance of two points. From the proposition, proved in Art. 20, it will follow that, if a system of straight lines in involution be cut by a transversal, the points of section will also be in involution.

33

CHAPTER II.

SPECIAL FORMS OF THE EQUATION OF THE SECOND DEGREE.

1. We now proceed to the discussion of the curve represented by the equation of the second degree. We shall first prove that every curve, represented by such an equation, is what is commonly called a conic section; and then, before proceeding further with the consideration of the general equation, shall investigate the nature of the curve corresponding to certain special forms of the equation.

PROP. Every curve represented by an equation of the second degree is cut by a straight line in two points, real, coincident, or imaginary.

The general equation of the second degree is represented by

uz2 + vß2 + wy2 + Qu'By + Qv′yx + 2w'aß = 0.

To find where the curve, of which this is the equation, is cut by the straight line

lx+mB+ny=0,

we may eliminate a between the two equations. This will give us a quadratic for the determination of

B

γ

to each of the

two values of this ratio, real, equal, or imaginary, one value of a will correspond; whence it appears that the straight line and the curve cut one another in two real, coincident, or imaginary points.

Hence, the curve is of the same nature as that represented by the equation of the second degree in Cartesian co-ordinates, and is, therefore, a conic section.

2. We shall now inquire what are the relations of the conic section to the triangle of reference, when certain relations exist among the coefficients of the equation.

First, suppose u, v, w, all = 0.

The equation then assumes the form

u'ẞy + v'yx + w'aß = 0,

which we shall write

λβγ + μγα + ναβ = 0.

Now, if in this equation we put a = 0, it reduces itself to λβγ = 0,

which requires either that ß=0, or that y=0.

It hence appears that the curve passes through two of the angular points (B, C) of the triangle of reference. It may similarly be shewn to pass through the third. Hence the equation

λβγ + μγα + ναβ = 0,

represents a conic, described about the triangle of reference.

3. Let us now inquire how the line

If in the equation of the conic we put + =0, or, .

ע

which is the same thing, μy+vß = 0, it reduces to λBy = 0. Hence the line + =0 meets the conic in the points in

βγ

μ

ע

which it meets the lines B=0, y = 0; but these two points coincide, since the line in question evidently passes through the point of intersection of B = 0 and y = 0. Hence the straight line and the conic meet one another in coincident points, that is, they touch one another at the point A.