ANHARMONIC RATIO.

23

ANHARMONIC RATIO.

19. We shall introduce, in this place, a short account of harmonic and anharmonic section, as a familiarity with this conception is useful in the higher geometrical investigations.

DEF. 1. If OP, OQ, OR, OS be four straight lines intersecting in a point, the ratio

sin POQ. sin ROS

sin POS. sin QOR

is called the anharmonic ratio of the pencil OP, OQ, OR, OS, and is expressed by the notation {0. PQRS}*.

DEF. 2. If P, Q, R, S be four points in a straight line, PQ.RS the ratio is called the anharmonic ratio of the range

PS. QR

P, Q, R, S, and may be expressed thus [PQRS].

In using these definitions, attention must be paid to the order in which the lines or points follow one another. Thus, the anharmonic ratio of the pencil OP, OR, OQ, OS, is different from that of the pencil OP, OQ, OR, OS, the former being equal to sin POR. sin QOS sin POQ. sin ROS sin POS. sin QOR'

the latter to

sin POS. sin QOR'

DEF. 3. If any number of straight lines, intersecting in a point, be cut by another straight line, the straight line which cuts the othersis called a transversal.

20. PROP. If four given straight lines, intersecting in a point O, be cut by a transversal in the points P, Q, R, S, the anharmonic ratio of the pencil OP, OQ, DR, OS, will be equal to that of the range P, Q, R, S.

* This notation is due, I believe, to Dr. Salmon. See his Conic Sections, p. 273 (third edition).

Thus the proposition is proved.

COR. 1. It appears, from the above proposition, that if a pencil be cut by two distinct transversals in P, Q, R, S and P, Q, R, S respectively, the anharmonic ratio of the range P, Q, R, S will be equal to that of the range P, Q, R, S', since each is equal to that of the pencil OP, OQ, OR, OS.

COR. 2. It appears also that, if four points P, Q, R, S, lying in a straight line, be joined with each of two other points O, O', the anharmonic ratios of the pencils OP, OQ,

HARMONIC PENCILS AND RANGES.

25

OR, OS; O'P, O'Q, O'R, 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.

For

=

PQ.RS PS. QR;

:. PQ (PS – PR) = PS (PR− PQ);

.. 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.

Fig. 10.

R

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.

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

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

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, ß−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.