On the Theory of Contours, and its Applications in Physical Science

1. In the first part of this paper we have considered merely the contours of curves, that is, contour points, and the method of obtaining the various physical diagrams. In this part we shall consider chiefly the contours of surfaces; that is, contour lines.

If any curve be cut by planes parallel to that of (x, y) and if the various points of intersection be projected on any one of these planes, say z = 0, the contour points so obtained will evidently lie on a definite line, and the line will be more accurately indicated in proportion as the number of intersecting planes is greater and their mutual distance is less. It will be given without any break in continuity by projecting every point of the curve upon the plane z = 0. But such a line may be regarded as the intersection, by the plane z = 0, (see fig. 48) of a cylindrical surface whose generating lines are parallel to the z-axis and are drawn from the given curve to meet that plane. We have here then the intersection of a given surface by a surface over which z is constant. But this satisfies our definition of a contour line. This case of a cylindrical surface supplies the simplest system of contour lines by giving z different values. The contours are all superposed in the diagram, but are not in general conterminous. The only case in which they would be conterminous is that in which the same values of the x and y co-ordinates of a point on the curve correspond to different values of the z-co-ordinate.