MR. K. PEARSON ON THE MATHEMATICAL THEORY OF EVOLUTION. 345 
fevers, in income tax and house duty returns, and in various types of anthropological 
measurements. It 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 : 
— Ce-??, 
to a form close to the normal frequency curve 
— (Cenez 
where C and p are constants. 
Hence any theory which is to cover the whole series of these curves must give a 
curve capable of varying from one to another of these types, @.e., from a type in 
which the maximum* practically coincides with the extreme ordinate, to a type in 
which it coincides with the central ordinate as in the normal frequency curve. 
It is well known that the points given by the point-binomial (4 + 4)” coincide very 
closely with the contour of a normal frequency curve when 7 is only moderately 
large. For example, the 21 points of (5 + $)” lie most closely on a normal frequency 
curve, and the author has devised a probability machine, which by continually bisecting 
streams of sand or rape seed for 20 successive falls gives a good normal frequency 
curve by the heights of the resulting 21 columns. Set to any other ratio p:q of 
division other than bisection, the machine gives the binomial (p + q)”, or indeed any 
less power and thus a wide range of asymmetrical point-binomials. Plate 7, fig. 2, 
represents, diagramatically, a 14-power binomial machine. 
Just as the normal frequency curve may be obtained by running a continuous 
curve through the point-binomial ( + 4)” when vn is fairly large, so a more general 
form of the probability curve may be obtained by running a continuous curve through 
the general binomial (p + q)”. As the great and only true test of the normal curve 
is: Does it really fit observations and measurements of a symmetrical kind ? so the 
best argument for the generalised probability curve deduced in this paper is that it 
does fit, and fit surprisingly accurately observations of an asymmetrical character. 
Indeed, there are very few results which have been represented by the normal curve 
which do not better fit the generalised probability curve,—a slight degree of 
asymmetry being probably characteristic of nearly all groups of measurements. 
Before deducing the generalised probability curve, it may be well to show how any 
asymmetrical curve may be fitted with its closest point-binomial. This will be the 
topic of the following five articles. 
(2.) Consider a series of rectangles on equal base c and whose heights are respec- 
tively the successive terms of the binomial (p+ q)" X a/c, where p+q=1. Here «is 
clearly the area of the entire system. Choose as origin a point O distant $c from the 
* J have found it convenient to use the term mode for the abscissa corresponding to the ordinate of 
maximum frequency. Thus the “mean,” the ‘‘ mode,” and the “median ” have all distinct characters 
important to the statistician. 
MDCCCXCV.—A. 2 
