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 = ƒs(x1, x2, ..., xn; a1, a2, ..., ar), (s = 1, 2, ..., n),
(i.)
where a1, a2, ..., ar represent the parameters. If this operation be represented by A, and that in which b1, b2, ..., br 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 = ƒs (x′1, x′2, ..., x′n; b1, b2, ..., br), (s = 1, 2, ..., n).
Since AB belongs to the group, the result of the elimination must be
x″s = ƒs (x1, x2, ..., xn; c1, c2, ..., cr),
where c1, c2, ..., cr 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, ..., xn must be
xs = ƒs (x′1, x′2, ..., x′n; d1, d2, ..., dr), (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.