MR. K. PEARSON ON THE MATHEMATICAL THEORY OF EVOLUTION. 381 
(20.) We have now considered methods for fully investigating whether a given 
system of measurements has a limited range, and for ascertaining the degree of 
skewness of the system. 
Analytically, our work may be expressed as follows :— 
The slope of the normal curve is given by a relation of the form 
The slope of the curve correlated to the skew binomial as the normal curve to the 
symmetrical binomial is given by a relation of the form 
LO SG 
y daz ~ ¢ + eg 
Finally, the slope of the curve correlated to the hypergeometrical series (which 
expresses a probability distribution in which the “contributory causes” are not 
independent, and not equally likely to give equal deviations in excess and defect) as 
the above curves to their respective binomials is given by a relation of the form 
1 dy _ — 2 

Co Ae 4b Gb OG 
This latter curve comprises the other two as special cases, and so far as my 
investigations have yet gone practically covers all homogeneous statistics that I have 
had to deal with. Something still more general may be conceivable, but I have 
hitherto found no necessity for it. 
To demonstrate its fitness and the importance of these generalised frequency 
distributions for various problems in physics, economics, and biology, I have devoted 
the remainder of this paper to the consideration of special cases of actual statistics. 
Part I[].—SraristicAL EXAMPLES. 
(21.) QuETELET, who often foreshadowed statistical advances without perceiving 
the method by which they might be scientifically dealt with, has treated of the subject 
of limits in Lettre XXII of his “ Lettres sur la Théorie des Probabilités” (1846). He 
seems to have been conscious that certain variations in excess or defect might 
biologically or physically be impossible, and he accordingly introduces the terms Limites 
extraordinaires en plus et en moms to mark the range of possible variation. He 
makes no attempt to show how this range may be found from a given set of statistics. 
“ Lorsqu’on suppose le nombre des observations infini, ou peut porter les écarts 4 des 
