and the different operations of the group will be given by different sets of values of the parameters a1, a2, ..., a_r. No one of these parameters is susceptible of continuous variations, but at least one must be capable of taking a number of values which is not finite, if the group is not one of finite order. Among the sets of values of the parameters there must be one which gives the identical transformation. No other transformation makes each of the differences x'1 - x1, x'2 - x2, ..., x'_n - x_n vanish. Let d be an arbitrary assigned positive quantity. Then if a transformation of the group can be found such that the modulus of each of these differences is less than d when the variables have arbitrary values within an assigned range of variation, however small d may be chosen, the group is said to be _improperly_ discontinuous. In the contrary case the group is called _properly_ discontinuous. The range within which the variables are allowed to vary may clearly affect the question whether a given group is properly or improperly discontinuous. For instance, the group defined by the equation x' = ax + b, where a and b are any rational numbers, is improperly discontinuous; and the group defined by x' = x + a, where a is an integer, is properly discontinuous, whatever the range of the variable. On the other hand, the group, to be later considered, defined by the equation x' = (ax + b)/(cx + d), where a, b, c, d are integers satisfying the relation ad - bc = 1, is properly discontinuous when x may take any complex value, and improperly discontinuous when the range of x is limited to real values. Linear discontinuous groups. Among the discontinuous groups that occur in analysis, a large number may be regarded as arising by imposing limitations on the range of variation of the parameters of continuous groups. If x'_s = [f]_s(x1, x2, ..., x_n; a1, a2, ..., a_r), (s = 1, 2, ..., n),