X. Contributions to the mathematical theory of evolution.—II. Skew variation in homogeneous material

Abstract An asymmetrical frequency curve may arise from two quite distinct classes of causes. In the first place the material measured may be heterogeneous and may consist of, a mixture of two or more homogeneous materials. Such frequency curves, for example, arise when we have a mixed population of two different races, a homogeneous population with a sprinkling of diseased or deformed members, a curve for tbe frequency of matrimony covering more than one class of the population, or in economics a frequency of interest curve for securities of different types of stabilit — railways and government stocks mixed with mining and financial companies. The treatment of this class of frequency curves requires us to break up the original curve into component parts, or simple frequency curves. This branch of the subject (for the special case of the compound being the sum of two normal curves) has been treated in a paper presented to the Poyal Society by the author, on October 18,1893. 1 lie second class of frequency curves arises in the case of homogeneous material when the tendency to deviation on one side of the mean is unequal to the tendency to deviation on the other side. Such curves arise in many physical, economic and biological investigations, for example, in frequency curves for the height of the barometer, in those for prices and for rates of interest of securities of the same class, in mortality curves, especially the percentage of deaths to cases in all kinds of fevers, in income tax and house duty returns, and in various types of anthropological measurements. I t is this class of curves, which are dealt with in the present paper. The general type of this class of frequency curve will be found to vary (see Plate 7, fig. 1) through all phases from the form close to the negative exponential curve: y = Ce-px, to a form close to the normal frequency curve y = Ce-px2 where C and p are constants.