x1 = f (x, y, z, a1, a2...)
x2 = φ (x, y, z, a1, a2...)
x3 = ψ (x, y, z, a1, a2...)
give an infinite family of real transformations of space, as to which we make the following hypotheses:
A. The functions f, φ, ψ, are analytical functions of
x, y, z, a1, a2....
B. Two points x1y1z1, x2y2z2 possess an invariant, i.e.
Ω(x1, y1, z1, x2, y2, z2) = Ω(x1′, y1′, z1′, x2′, y2′, z2′)
where x1′..., x2′..., are the transformed coordinates of the two points.
C. Free Mobility: i.e., any point can be moved into any other position; when one point is fixed, any other point of general position can take up ∞2 positions; when two points are fixed, any other of general position can take up ∞1 positions; when three, no motion is possible—these limitations being results of the equations given by the invariant Ω.
[64] On this point, cf. Klein, Höhere Geometrie, Göttingen, 1893, II. pp. 225–244, especially pp. 230–1.
[65] Axiom II. of the metrical triad corresponds to Axiom III. of the projective, and vice versâ.