(AB, CD) = (CD, AB) = (BA, DC) = (DC, BA).
II. If in a cross-ratio the two points belonging to one of the two groups be interchanged, the cross-ratio changes into its reciprocal, i.e.
(AB, CD) = 1/(AB, DC) = 1/(BA, CD) = 1/(CD, BA) = 1/(DC, AB).
From I. and II. we see that eight cross-ratios are associated with (AB, CD).
III. If in a cross-ratio the two middle letters be interchanged, the cross-ratio α changes into its complement 1 − α, i.e. (AB, CD) = 1 − (AC, BD).
[§ 16. If λ = (AB, CD), μ = (AC, DB), ν = (AD, BC), then λ, μ, ν and their reciprocals 1/λ, 1/μ, 1/ν are the values of the total number of twenty-four cross-ratios. Moreover, λ, μ, ν are connected by the relations
λ + 1/μ = μ + 1/ν = ν + 1/λ = −λμν = 1;
this proposition may be proved by substituting for λ, μ, ν and reducing to a common origin. There are therefore four equations between three unknowns; hence if one cross-ratio be given, the remaining twenty-three are determinate. Moreover, two of the quantities λ, μ, ν are positive, and the remaining one negative.
The following scheme shows the twenty-four cross-ratios expressed in terms of λ, μ, ν.]
| (AB, CD) (BA, DC) (CD, AB) (DC, BA) | λ | 1 − μ | 1/(1 − ν) | (AD, BC) (BC, AD) (CB, DA) (DA, CB) | (λ − 1)/λ | μ/(μ − 1) | ν |
| (AC, DB) (BD, CA) (CA, BD) (DB, AC) | 1/(1 − λ) | 1/μ | (ν − 1)/ν | (AC, BD) (BD, AC) (CA, DB) (DB, CA) | 1 − λ | μ | ν/(ν − 1) |
| (AB, DC) (BA, CD) (CD, BA) (DC, AB) | 1/λ | 1/(1 − μ) | 1 − ν | (AD, CB) (BC, DA) (CB, AD) (DA, BC) | λ/(λ − 1) | (μ − 1)/μ | 1/ν |