taken as infinitesimal operations from which to generate a continuous
group among the infinitesimal operations of the group, there must
occur the combinant of x2ddx and x3ddx. This is x4ddx. The combinant
of this and x2ddx is 2x5ddx and so on. Hence xrddx, where r is any
positive integer, is an infinitesimal operation of the group. The
general infinitesimal operation of the group is therefore ƒ(x)ddx, where
ƒ(x) is an arbitrary integral function of x.
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 ƒ(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.
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 Contact transformations. 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
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
| x′ = x + | pt | , y ′ = y + | qt | , z′ = z − | t | , |
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.
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 whichApplications of the theory of continuous groups. 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 ƒ = a0xn + na1xn−1y + . . . + any n, the variables be subjected to a homogeneous substitution
| x′ = αx + βy, y ′ = γx + δy, | (i.) |
and if the coefficients in the new form be represented by accenting the old coefficients, then
| a′0 = a0αn + a1nαn−1γ + . . . + anγn, a′1 = a0αn−1β + a1 {(n−1) αn−2βγ + αn−1δ} + . . . + anγn−1δ, · · · · · · a′n = a0βn + a1nβn−1δ + . . . + anδn; |
and this is a homogeneous linear substitution performed on the coefficients. The totality of the substitutions, (i.), for which αδ − βγ = 1, constitutes a continuous group of order 3, which is generated by the two infinitesimal transformations y(∂/∂x) and x(∂/∂y ). Hence with
