In the classification of the groups, projective or non-projective of two or more variables, the distinction between primitive and imprimitive groups immediately presents itself. For groups of the plane the following question arises. Is there or is there not a singly-infinite family of curves [f](x, y) = C, where C is an arbitrary constant such that every operation of the group interchanges the curves of the family among themselves? In accordance with the previously given definition of imprimitivity, the group is called imprimitive or primitive according as such a set exists or not. In space of three dimensions there are two possibilities; namely, there may either be a singly infinite system of surfaces F(x, y, z) = C, which are interchanged among themselves by the operations of the group; or there may be a doubly-infinite system of curves G(x, y, z) = a, H(x, y, z) = b, which are so interchanged. In regard to primitive groups Lie has shown that any primitive group of the plane can, by a suitably chosen transformation, be transformed into one of three definite types of projective groups; and that any primitive group of space of three dimensions can be transformed into one of eight definite types, which, however, cannot all be represented as projective groups in three dimensions. The results which have been arrived at for imprimitive groups in two and three variables do not admit of any such simple statement. Contact transformations. We shall now explain the conception of contact-transformations and groups of contact-transformations. This conception, like that of continuous groups, owes its origin to Lie. From a purely analytical point of view a contact-transformation may be defined as a point-transformation in 2n + 1 variables, z, x1, x2, ..., x_n, p1, p2, ..., p_n which leaves unaltered the equation dz - p1dx1 - p2dx2 - ... - p_ndx_n = 0. Such a definition as this, however, gives no direct clue to the geometrical properties of the transformation, nor does it explain the name given.