1. From now on the part of the group connected to the identity element will be understood unless stated otherwise. Thus we exclude improper transformations like reflections, which also preserve the line element (1).
2. The proofs may be found in old-fashioned books on «higher geometry» likeF. Klein:Vorlesungen über höhere Geometrie, 3rd ed. (Berlin, 1926). Theorem (6) is a combination of several theorems, only one of which is historically called Liouville's theorem.
3. For at least some kinds of force.
4. See ref. (16) SF II, sect.3, of ref. (13) of part I.
5. AllX a withX n+2=0 are lumped into a «point at infinity» in the projective sense. Note, however, that both spheres (x μ, λ) and (x μ, −λ) are mapped into the same projective point. Ignore this doubling for the moment.