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, [oo]^5 surface elements in three-dimensional space. The surface-elements of a surface form a system of [oo]^2 elements, for there are [oo]^2 points on the surface, and at each a definite surface-element. The surface-elements of a curve form, again, a system of [oo]^2 elements, for there are [oo]^1 points on the curve, and at each [oo]^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 [oo]^2 elements. Now each of these systems of [oo]^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 [oo]^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.