OR, OS; O'P, O'Q, OʻR, O'S, will be equal to one another, since each is equal to that of the range P, Q, R, S.

21. DEF. A pencil, of which the anharmonic ratio is unity, is called an harmonic pencil.

A range, of which the anharmonic ratio is unity, is called an harmonic range, and the straight line, on which the range lies, is said to be divided harmonically.

From what has been said above, it will be seen that, if an harmonic pencil be cut by a transversal, the four points of section will form an harmonic range. And if four points, forming an harmonic range, be joined with a fifth point, the four joining lines will form an harmonic pencil.

The line OS is said to be a fourth harmonic to the pencil OP, OQ, OR; and the point S to be a fourth harmonic to the range P, Q, R.

The term harmonic is employed on account of the circumstance, that if the points P, Q, R, S form what is above defined as an harmonic range, PR will be an harmonic mean between PQ and PS.

.. PQ: PS:: PR-PQ: PS-PR,

whence PQ, PR, PS are in harmonical progression.

From the above proportion it appears that if PQ = QR, PS. Hence, if PR be bisected in Q, the fourth harmonic to the range P, Q, R is infinitely distant. Or, as it may otherwise be stated, if PR be bisected in Q, and P, Q, R be joined with any point O, not in the line PR, the fourth harmonic to the pencil OP, OQ, OR, will be parallel to the transversal PQR.

22. PROP. The external and internal bisectors of any angle form, with the lines containing the angle, an harmonic pencil.

Let the angle POR be bisected internally by OQ, let PO be produced to any point P', and let the angle P'OR be bisected by OS, then

23. PROP. If ABC be the triangle of reference, and AD, AE straight lines respectively represented by the equations

k'

B-ky =0, B+k'y = 0,

then will be the anharmonic ratio of the pencil AB, CA,

k

AD, AE.

Fig. 11.

B

E

Let BC cut AD, AE respectively in D, E, then since D is a point in the line ẞ-ky = 0,

and since E is a point in the line ẞ+k'y = 0,

BE AABE су с

=

=

CE

AACE - bß

== ;
bk

BD.CE k

or

k'

k

is the anharmonic ratio of the range B, D, C, E; that

is, of the pencil AB, AD, AC, AE.

COR. It hence follows that the straight lines respectively represented by the equations B=0, B-ky=0,y=0, B+ky=0, form an harmonic pencil.

24. Hence we deduce a geometrical construction for the determination of the fourth harmonic to three given intersecting straight lines.

Let AB, AD, AC be three given intersecting straight lines, and let it be required to find a straight line AE, such that AB, AD, AC, AE shall form an harmonic pencil.

B

Fig. 12.

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 AẾ 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 λα + β - 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. point; and AD, sect BC in D;

If ABC be a given triangle, P any given the fourth harmonic to AB, AP, AC, interBE, the fourth harmonic to BC, BP, BA,

intersect CA in E; CF, the fourth harmonic to CA, CP, CB, intersect AB in F; 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