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.
We shall now explain the conception of contact-transformations and groups of contact-transformations. This conception, Contact transformations. 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, ..., xn, p1, p2, ..., pn which leaves unaltered the equation dz − p1dx1 − p2dx2 − ... − pndxn = 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.
In dealing with contact-transformations we shall restrict ourselves to space of two or of three dimensions; and it will be necessary to begin with some purely geometrical considerations. An infinitesimal surface-element in space of three dimensions is completely specified, apart from its size, by its position and orientation. If x, y, z are the co-ordinates of some one point of the element, and if p, q, −1 give the ratios of the direction-cosines of its normal, x, y, z, p, q are five quantities which completely specify the element. There are, therefore, ∞5 surface elements in three-dimensional space. The surface-elements of a surface form a system of ∞2 elements, for there are ∞2 points on the surface, and at each a definite surface-element. The surface-elements of a curve form, again, a system of ∞2 elements, for there are ∞1 points on the curve, and at each ∞1 surface-elements containing the tangent to the curve at the point. Similarly the surface-elements which contain a given point clearly form a system of ∞2 elements. Now each of these systems of ∞2 surface-elements has the property that if (x, y, z, p, q) and (x + dx, y + dy, z + dz, p + dp, q + dq) are consecutive elements from any one of them, then dz − pdx − qdy = 0. In fact, for a system of the first kind dx, dy, dz are proportional to the direction-cosines of a tangent line at a point of the surface, and p, q, −1 are proportional to the direction-cosines of the normal. For a system of the second kind dx, dy, dz are proportional to the direction-cosines of a tangent to the curve, and p, q, −1 give the direction-cosines of the normal to a plane touching the curve; and for a system of the third kind dx, dy, dz are zero. Now the most general way in which a system of ∞2 surface-elements can be given is by three independent equations between x, y, z, p and q. If these equations do not contain p, q, they determine one or more (a finite number in any case) points in space, and the system of surface-elements consists of the elements containing these points; i.e. it consists of one or more systems of the third kind.
If the equations are such that two distinct equations independent of p and q can be derived from them, the points of the system of surface-elements lie on a curve. For such a system the equation dz − pdx − qdy = 0 will hold for each two consecutive elements only when the plane of each element touches the curve at its own point.
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 ∞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′ = ƒ1(x, y, z, p, q), ..., q′ = ƒ5(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 ∞2 elements into another system of ∞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 ∞1 line-elements, which belong to a curve, into ∞1 line-elements which again belong to a curve; while two curves which touch are transformed into two other curves which touch.
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 x2 + y2 − 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