Among the groups isomorphic with a given continuous group there is one of special importance which is known as the _adjunct_ group. This is a homogeneous linear group in a number of variables equal to the order of the group, whose infinitesimal operations are defined by the relations [Pd] X_i=[Sigma] c_(ijs)x_i -------, (j = 1, 2, ..., r), i, s [Pd]x_s where c_(ijs) are the often-used constants, which give the combinants of the infinitesimal operations in terms of the infinitesimal operations themselves. That the r infinitesimal operations thus defined actually generate a group isomorphic with the given group is verified by forming their combinants. It is thus found that (X_pX_q) = [Sigma][s]c_(pqs)X_s. The X's, however, are not necessarily linearly independent. In fact, the sufficient condition that [Sigma][j]a_jX_j should be identically zero is that [Sigma][j]a_jc_(ijs) should vanish for all values of i and s. Hence if the equations [Sigma][j]a_jc_(ijs) = 0 for all values of i and s have r' linearly independent solutions, only r - r' of the X's are linearly independent, and the isomorphism of the two groups is multiple. If Y1, Y2, ..., Y_r are the infinitesimal operations of the given group, the equations [Sigma] a_jc_(ijs) = 0, (s, i = 1, 2, ..., r) j express the condition that the operations of the cyclical group generated by [Sigma][j]a_jY_i should be permutable with every operation of the group; in other words, that they should be self-conjugate operations. In the case supposed, therefore, the given group contains a subgroup of order r' each of whose operations is self-conjugate. The adjunct group of a given group will therefore be simply isomorphic with the group, unless the latter contains self-conjugate operations; and when this is the case the order of the adjunct will be less than that of the given group by the order of the subgroup formed of the self-conjugate operations. Continuous groups of the line of the plane, and of three-dimensional space. We have been thus far mainly concerned with the abstract theory of continuous groups, in which no distinction is made between two simply isomorphic groups. We proceed to discuss the classification and theory of groups when their form is regarded as essential; and this is a return to a more geometrical point of view. It is natural to begin with the projective groups, which are the simplest in form and at the same time are of supreme importance in geometry. The general projective group of the straight line is the group of order three given by ax + b x' = ------- cx + d' where the parameters are the ratios of a, b, c, d. Since x'3 - x'2 x' - x'1 x3 - x2 x - x1 --------- . -------- = ------- . ------ x'3 - x'1 x' - x'2 x3 - x1 x - x2