x"_s = [f]_s(x1, x2, ..., x_n; c1, c2, ..., c_r),
where c1, c2, ..., c_r represent another definite set of values of the
parameters. Moreover, since A^(-1) belongs to the group, the result of
solving equations (i.) with respect to x1, x2, ..., x_n must be
x_s = [f]_s(x'1, x'2, ..., x'_n; d1, d2, ..., d_r),
(s = 1, 2, ..., n).
Conversely, if equations (i.) are such that these two conditions are
satisfied, they do in fact define a finite continuous group.
Infinitesimal operation of a continuous group.
It will be assumed that the r parameters which enter in equations (i.)
are independent, i.e. that it is impossible to choose r' (< r)
quantities in terms of which a1, a2, ..., a_r can be expressed. Where
this is the case the group will be spoken of as a "group of order r."
Lie uses the term "_r-gliedrige Gruppe_." It is to be noticed that the
word order is used in quite a different sense from that given to it in
connexion with groups of finite order.
In regard to equations (i.), which define the general operation of the
group, it is to be noticed that, since the group contains the
identical operation, these equations must for some definite set of
values of the parameters reduce to x'1 = x1, x'2 = x2, ..., x'_n =
x_n. This set of values may, without loss of generality, be assumed to
be simultaneous zero values. For if i1, i2, ..., i_r be the values of
the parameters which give the identical operation, and if we write
a_s = i_s + a, (s = 1, 2, ..., r),
then zero values of the new parameters a1, a2, ..., a_r give the
identical operation.
To infinitesimal values of the parameters, thus chosen, will
correspond operations which cause an infinitesimal change in each of
the variables. These are called infinitesimal operations. The most
general infinitesimal operation of the group is that given by the
system
[Pd][f]_s [Pd][f]_s [Pd][f]_s
x'_s - x_s = [delta]x_s = --------- [delta]a1 + --------- [delta]a2 + ... + --------- [delta]a_r, (s = 1, 2, ..., n),
[Pd]a1 [Pd]a2 [Pd]a_r
where, in [Pd][f]_s/[Pd]a_i, zero values of the parameters are to be
taken. Since a1, a2, ..., a_r are independent, the ratios of
[delta]a1, [delta]a2, ..., [delta]a_r are arbitrary. Hence the most
general infinitesimal operation of the group may be written in the
form
/ [Pd][f]_s [Pd][f]_s [Pd][f]_s\
[delta]x_s = ( e1--------- + e2--------- + ... + e_r--------- ) [delta]t, (s = 1, 2, ..., n),
\ [Pd]a1 [Pd]a2 [Pd]a_r /
where e1, e2, ..., e_r are arbitrary constants, and [delta]t is an
infinitesimal.
If F(x1, x2, ..., x_n) is any function of the variables, and if an
infinitesimal operation of the group be carried out on the variables
in F, the resulting increment of F will be
[Pd]F [Pd]F [Pd]F
------[delta]x1 + ------[delta]x2 + ... + -------[delta]x_n.
[Pd]x1 [Pd]x2 [Pd]x_n