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 If the differential operator [Pd][f]1 [Pd] [Pd][f]2 [Pd] [Pd][f]_n [Pd] -------- ------ + -------- ------ + ... + --------- ------- [Pd]a_i [Pd]x1 [Pd]a_i [Pd]x2 [Pd]a_i [Pd]x_n be represented by X_i, (i = 1, 2, ..., r), then the increment of F is given by (e1X1 + e2X2 + ... + e_rX_r)F[delta]t.