A Useful Empirical Formula

Abstract

MY note in NATURE of October 8 may be extended. I found that one of my pupils, Mr. Glasgow, was assuming that the expansion and compression parts of a gas engine diagram followed laws of the type pvⁿ constant; in cases where there was a probability that the clearance had not been measured accurately, so that v being the measured volume and c the constant error, he assumed p(v+e)ⁿ to be constaznt, and he was enabled to find c from the curve. There is no reason to believe that these curves ought to have such a law, although, curiously enough, following this assumption, the clearance obtained from the compression curve is usually not very different from that obtained from the expansion curve. Mr. Glasgow's method of finding c is much the same as what I shall now describe. An empirical formula of the type y=a+bxⁿ would be exceedingly useful in many parts of pure and applied science if, when given a table of values of y and x, we could readily find a, b and n. I have often sought for a method of working, but without success. If a is zero, we have only to plot log y and log x as the coordinates of points on squared paper. If a is not zero, there is a clumsy method of using logarithmic paper which may be adopted, but it is not satisfactory. We now have a method easy of application. Thus values of x and y being given, draw the curve AB shown in the figure. Set off any convenient angle DOX. Select the point P. Draw PD, XF, FEQ, EH, &c, the lines XF, EH, &c, being at 45°. Project horizontally from the points PQR, &c, to M, U or N, W or V, &c, letting lines at 45° from M, U, &c, meet the horizontals at N, V, X, &c. If the above law holds, N, V, X, Z lie in a straight line. If they lie only approximately in a straight line, draw the line N′O′ lying most evenly among them. Then OO′ is the value of a, and n is log (1+tan N′O′Y)/log (1+tan DOX), and b is readily found.