Fig. 12. B D C Through D, any point of the second of the three given straight lines, draw two transversals BDC, B'DC', cutting AB in B, B′, AC in C, C' respectively. Join B'C, BC', and produce them to meet in E. Join AE, then AE shall be the fourth harmonic required. For, let ABC be the triangle of reference, and let the equation of AD be B-ky = 0. Let the equation of B'C' be λa+B―ky = 0. Then that of BC' is λa-ky =0, B' C ...λα + β = 0, AE B+ ky =0, ... whence AE is the fourth harmonic required. 25. PROP. If ABC be a given triangle, P any given point; and AD, the fourth harmonic to AB, AP, AC intersect BC in D; BE, the fourth harmonic to BC, BP, BA intersect CA in E; intersect AB in F; HARMONIC PENCILS. 29 CF, the fourth harmonic to CA, CP, CB, then D, E, F lie in the same straight line. Let f, g, h be the co-ordinates of P. Then the equation of AP is From the form of these equations it will be seen that 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, 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 OP. OP'OQ.OQ' OR. OR'...... = Then these points are said to form a system in involution. If K be a point such that OK2=k, 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 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. Then, by definition, (p − x) (p'— x) = (q − x) (q' − x) ; pp'-qq' which determines the centre. 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. Ρ k — and K,P. K,P = (k − *) (* + p) = 1⁄2) (p2 − k2) ; .. K1P.KP = K1P'. K„P, Ρ 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. |