§ 17. If one of the points of which a cross-ratio is formed is the point at infinity in the line, the cross-ratio changes into a simple ratio. It is convenient to let the point at infinity occupy the last place in the symbolic expression for the cross-ratio. Thus if I is a point at infinity, we have (AB, CI) = −AC/CB, because AI : IB = −1.
Every common ratio of three points in a line may thus be expressed as a cross-ratio, by adding the point at infinity to the group of points.
Harmonic Ranges
§ 18. If the points have special positions, the cross-ratios may have such a value that, of the six different ones, two and two become equal. If the first two shall be equal, we get λ = 1/λ, or λ² = 1, λ = ±1.
If we take λ = +1, we have (AB, CD) = 1, or AC/CB = AD/DB; that is, the points C and D coincide, provided that A and B are different.
If we take λ = −1, so that (AB, CD) = −1, we have AC/CB = −AD/DB. Hence C and D divide AB internally and externally in the same ratio.
The four points are in this case said to be harmonic points, and C and D are said to be harmonic conjugates with regard to A and B.
But we have also (CD, AB) = −1, so that A and B are harmonic conjugates with regard to C and D.
The principal property of harmonic points is that their cross-ratio remains unaltered if we interchange the two points belonging to one pair, viz.
(AB, CD) = (AB, DC) = (BA, CD).