F. R. MACAULAY' and other writers2 have noted a tendency for changes in the square roots of common-stock prices to be constant regardless of price level. This phenomenon has naturally suggested that when such prices are represented graphically the charts be designed so that vertical distances from the origin are proportional to the square roots of the prices indicated in the margins. There is some reason to believe that charts of this kind might also be useful in plotting other kinds of data. Assume that n sales are distributed at random over 1/p firms during some interval of time, and that u1, u2, *, u1/, are the actual numbers of sales made by the different firms F1, F2, , Fil, Then, a priori, the probability that a particular firm will make one of these sales is p, and the mean and variance of the u's will tend to be

This article focuses on the prudent investment theory in public utility rate making. It is author's opinion that successful regulation of public utility rates cannot be accomplished under the fair-value doctrine and that the investment method must be sanctioned if justice is to be done to the consumer, the utility, and the general public as well. Stated somewhat differently the author believes the fair-value basis of rate making altogether impracticable and unworkable, that it is basically wrong in its economic concept, that the circumstances which gave birth to the principle have long since ceased to exist, and that is a reasonably good job of public utility rate regulation is to be achieved it is through investment approach. No review of rate regulatory procedures in this country would be complete without a brief reference to leading decisions of the Supreme Court of the U.S. on the subject. Not only did the fair-value doctrine, which plagued regulation for many years, have its real genesis in a decision of that Court, but the decisions of that body have greatly influenced the thinking and pretty well dominated the practices in respect to public utility rate regulation.

IF the variable X represents the price of an active stock, it is well known that the tendency of X to change is an increasing function of price itself, say f(X). If the tendency to relative change with respect to price were constant, graphs whose amplitude of fluctuation is uncorrelated with price could be constructed by (1) plotting log X on ordinary arithmetic scale or (2) plotting X on semilogarithmic scale. However, the tendency to relative change f(X)/X has been found to be a decreasing function of X and hence any constant-amplitude scale for plotting stock prices must be based on some function other than log X, say F(X), whose derivative is inversely proportional to the tendency to change with respect to price. In other words, the desired function is any solution of the differential equation