One of the simplest instances of a contact-transformation that can be given is the transformation by reciprocal polars. By this transformation a point P and a plane p passing through it are changed into a plane p' and a point P' upon it; i.e. the surface-element defined by P, p is changed into a definite surface-element defined by P', p'. The totality of surface-elements which belong to a (non-developable) surface is known from geometrical considerations to be changed into the totality which belongs to another (non-developable) surface. On the other hand, the totality of the surface-elements which belong to a curve is changed into another set which belong to a developable. The analytical formulae for this transformation, when the reciprocation is effected with respect to the paraboloid x^2 + y^2 - 2z = 0, are x' = p, y' = q, z' = px + qy - z, p' = x, q' = y. That this is, in fact, a contact-transformation is verified directly by noticing that dz' - p'dx' - q'dy' = -d(z - px - qy) - xdp - ydq = -(dz - pdx - qdy). A second simple example is that in which every surface-element is displaced, without change of orientation, normal to itself through a constant distance t. The analytical equations in this case are easily found in the form pt qt x' = x + ---------------------, y' = y + --------------------, [root](1 + p^2 + q^2) [root](1 + p^2 + q^2) t z' = z - ---------------------, [root](1 + p^2 + q^2) p' = q, q' = q. That this is a contact-transformation is seen geometrically by noticing that it changes a surface into a parallel surface. Every point is changed by it into a sphere of radius t, and when t is regarded as a parameter the equations define a cyclical group of contact-transformations. The formal theory of continuous groups of contact-transformations is, of course, in no way distinct from the formal theory of continuous groups in general. On what may be called the geometrical side, the theory of groups of contact-transformations has been developed with very considerable detail in the second volume of Lie-Engel. Applications of the theory of continuous groups. To the manifold applications of the theory of continuous groups in various branches of pure and applied mathematics it is impossible here to refer in any detail. It must suffice to indicate a few of them very briefly. In some of the older theories a new point of view is obtained which presents the results in a fresh light, and suggests the natural generalization. As an example, the theory of the invariants of a binary form may be considered. If in the form [f] = a0x^_n + na1x^(n-1)y + ... + a_ny^n, the variables be subjected to a homogeneous substitution x' = [alpha]x + [beta]y, y' = [gamma]x + [delta]y, (i.)