If the equations are such that only one equation independent of p and q can be derived from them, the points of the system of surface-elements lie on a surface. Again, for such a system the equation dz - pdx - qdy = 0 will hold for each two consecutive elements only when each element touches the surface at its own point. Hence, when all possible systems of [oo]^2 surface-elements in space are considered, the equation dz - pdx - qdy = 0 is characteristic of the three special types in which the elements belong, in the sense explained above, to a point or a curve or a surface. Let us consider now the geometrical bearing of any transformation x' = [f]1(x, y, z, p, q), ..., q' = [f]5(x, y, z, p, q), of the five variables. It will interchange the surface-elements of space among themselves, and will change any system of [oo]^2 elements into another system of [oo]^2 elements. A special system, i.e. a system which belongs to a point, curve or surface, will not, however, in general be changed into another special system. The necessary and sufficient condition that a special system should always be changed into a special system is that the equation dz' - p'dx' - q'dy' = 0 should be a consequence of the equation dz - pdx - qdy = 0; or, in other words, that this latter equation should be invariant for the transformation. When this condition is satisfied the transformation is such as to change the surface-elements of a surface in general into surface-elements of a surface, though in particular cases they may become the surface-elements of a curve or point; and similar statements may be made with respect to a curve or point. The transformation is therefore a veritable geometrical transformation in space of three dimensions. Moreover, two special systems of surface-elements which have an element in common are transformed into two new special systems with an element in common. Hence two curves or surfaces which touch each other are transformed into two new curves or surfaces which touch each other. It is this property which leads to the transformations in question being called contact-transformations. It will be noticed that an ordinary point-transformation is always a contact-transformation, but that a contact-transformation (in space of n dimensions) is not in general a point-transformation (in space of n dimensions), though it may always be regarded as a point-transformation in space of 2n + 1 dimensions. In the analogous theory for space of two dimensions a line-element, defined by (x, y, p), where 1 : p gives the direction-cosines of the line, takes the place of the surface-element; and a transformation of x, y and p which leaves the equation dy - pdx = 0 unchanged transforms the [oo]^1 line-elements, which belong to a curve, into [oo]^1 line-elements which again belong to a curve; while two curves which touch are transformed into two other curves which touch.