Note 63 — Chapter I · A SHORT HISTORY OF METAGEOMETRY.
Reading edition
Lie's premisses, to be accurate, are the following:
Let
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.