The object of the following paper is to illustrate how readily some problems, which frequently occur in physical work, may be solved by the application of certain methods which are not very generally employed in mathematical investigations. The method on which the following proof is based is that of contour lines. These are a device enabling us to represent the third dimension on a plane, and are such that all the points on one contour line are at one and the same height above the level of the base plane. Probably the best known application of the principle of contours is to be found in the Survey Maps, where successive (closed) curves pass through all points at heights of 100, 200, 300, &c., feet above the sea-level; but they are of equal value in many physical diagrams, where they appear as equipotential lines, isothermals, lines of equal illumination, lines of equal force in a magnetic field, as well as in many other forms. In this paper they are employed as representing the third dimension merely. When contour lines are placed so as to indicate an equal rise in the intervals between each successive pair [or, in fact, according to any definite system], we can, with their aid, tell two things about a surface; the one—what is the steepness, or gradient, in any direction from a given point; the other—which is the direction of steepest slope at any point. The absolute steepness is measured by the amount of vertical motion per unit of horizontal motion of a point; i.e., by the tangent of the angle between the horizontal plane and the line in which the point moves. This is a fact with which everyone is acquainted ; every person knows what is meant by saying that the gradient along a line of railway is 1 in 50 or 1 in 500, as the case may be. To find the steepness at any point we merely need to know, in terms of the length of the line taken to represent unit steepness, the length of the line passing through the point in the given direction, and terminated by the two contours on either side of that point. The length of the line will then be inversely as the steepness.