Rate of Change as Determined Graphically with an Equilateral Glass Prism

RATE OF CHANGE AS DETERMINED GRAPHICALLY WITH AN EQUILATERAL GLASS PRISM ON CERTAIN OCCASIONS an investigator may wish to evaluate the rate of change of an experimental variable with respect to another, but finds it not convenient or feasible to measure the rate of change directly; instead, only cumulative values of the two variables are available. The authors encountered this situation in sand-model studies of water infiltration (2, 3, 4) where the cumulative volume of water adsorbed was recorded as a function of total elapsed time. The plot of water volume versus time was curvilinear; hence, obtaining water flow rates involved graphical differentiation of the curve at various points. This was accomplished by the use of an equilateral glass prism, a procedure first encountered by the second author in a physical chemistry course at the University of Maryland. Since knowledge of the procedure does not appear to be widespread, the purpose of this note is to describe the method as developed by the authors, and to evaluate its accuracy. Details of the method are discussed with reference to figure 1. With the apex of its equilateral section directly above the point P, the prism is placed across the curve MPQ. If the alignment of the prism is not perpendicular to the curve at P, the curve image in the prism will appear to have a distinct discontinuity at the prism apex. So, the prism is rotated about the point P until the curve appears to pass continuously through the prism. The straight-line secant O'S' is drawn along the side of the prism with a sharp pencil. In actual practice, the straight line OPS is not drawn, even though it is tangent to the curve at P and defines the geometric slope. One reason for this is that the prism method yields the secant O'S' naturally. Secondly, in a series of differentiations at narrowly spaced intervals along the curve, the various tangent lines would overlap confusingly. In the present instance, OPS is shown simply to demonstrate the calculations. Assume now that the geometric length-measuring scales on both abscissa and ordinate are identical. If centimeter graph paper is being used, the logical geometric unit length is 1 cm. However, the units of x can be something else (such as minutes), and the units of y can be still different (such as ml.). Let Nx be the number of x-units per unit geometric scale length on the x-axis, and Ny the number of y-units per unit geometric scale length on the yaxis. Let "a" be the geometric length of the altitude of the slope-defining right triangle for OPS, while b is the geometric length of the base of this triangle. Then, Ay = 2 — Yi = Ny> Ax = x2 — xt = Nxb, and R, die rate of change of y with x at the point P, is