The determination of a particular operation of a given continuous group depends on assigning special values to each one of a set of parameters which are capable of continuous variation. The first distinction regards the number of these parameters. If this number is finite, the group is called a _finite_ continuous group; if infinite, it is called an _infinite_ continuous group. In the latter case arbitrary functions must appear in the equations defining the operations of the group when these are reduced to an analytical form. The theory of infinite continuous groups is not yet so completely developed as that of finite continuous groups. The latter theory will mainly occupy us here. Sophus Lie, to whom the foundation and a great part of the development of the theory of continuous groups are due, undoubtedly approached the subject from a geometrical standpoint. His conception of an operation is to regard it as a geometrical transformation, by means of which each point of (n-dimensional) space is changed into some other definite point. The representation of such a transformation in analytical form involves a system of equations, x'_s = [f]_s(x1, x2, ..., x_n), (s = 1, 2, ..., n), expressing x'1, x'2, ..., x'_n, the co-ordinates of the transformed point in terms of x1, x2, ..., x_n, the co-ordinates of the original point. In these equations the functions [f]_s are analytical functions of their arguments. Within a properly limited region they must be one-valued, and the equations must admit a unique solution with respect to x1, x2, ..., x_n, since the operation would not otherwise be a definite one. From this point of view the operations of a continuous group, which depends on a set of r parameters, will be defined analytically by a system of equations of the form x'_s = [f]_s(x1, x2, ..., x_n; a1, a2, ..., a_r), (s = 1, 2, ..., n), (i.) where a1, a2, ..., a_r represent the parameters. If this operation be represented by A, and that in which b1, b2, ..., b_r are the parameters by B, then the operation AB is represented by the elimination (assumed to be possible) of x'1, x'2, ..., x'_n between the equations (i.) and the equations x"_s = [f]_s(x'1, x'2, ..., x'_n; b1, b2, ..., b_r), (s = 1, 2, ..., n). Since AB belongs to the group, the result of the elimination must be 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.