The axioms of descriptive geometry, by A.N. Whitehead.

46

LIEIS FIRST SOLUTION

[CH. V

to some extent faulty. For they implicitly assume that there is one
definite group of motions, as indeed our sensations of the physical
world do in fact seem to give us special intelligence of one such definite
group in physical space. However it will be found that an indefinite
number of groups of one-one point transformations exist which satisfy
Lie's definitions of the properties of a complete group of motions.
Accordingly a motion when one special group is being considered is not
a motion when another such group is considered.
A group of motions is called a congruence-group, and the
definitions of the characteristics of such a group are called the axioms
of Congruence.
43. Lie's results, as expressed by himself, are as follows:
Definition*. A finite continuous group in the variables x,, x,,... x
is called transitive, if in the space (xi, x,,...x) an n-fold extended
region exists, within which each point can be transformed into any
other point through at least one transformation of the group.
Definitiont. A real continuous group of three-fold extended
space possesses at the real point P free mobility in the infinitesimal,
if it satisfies the following conditions: If a point P and an arbitrary
real line-element passing through it are fixed, continuous motion is still
possible; but if, in addition to P and that line-element, an arbitrary
real surface-element, passing through both is held fixed, then shall no
continuous motion be further possible.
Theorem +. (1) If a real continuous projective group of ordinary
three-fold extended space possesses without exception in all real points
of this space free mobility in the infinitesimal, then it is six-limbed and
transitive, and consists of all real projective transformations through
which a not-exceptional imaginary surface of the second degree, which
is represented by a real equation [e.g. x + y2 + z2 + 1 = 0], remains
invariant (latent).
(2) If a real continuous projective group of ordinary three-fold
extended space possesses free mobility in the infinitesimal, not in all
real points of this space but only in all real points of a certain region,
then it is six-limbed and transitive and is either the continuous real
projective group of a not-exceptional real not-ruled surface of the
* Cf. Theorie der Transformationsgruppen, vol. I. ~ 58.
t Cf. loc. cit., vol. III. ~ 98.
+ Cf. Lie, loc. cit. vol. IIL ~ 98.