I. Introduction. 



Among newer investigations into the theory of probability I know none more 

 important than those of Pearson in his admirable series of »Contrilnüions to the 

 mathematical theory of evolution». The numerous school of biologists that has 

 grown up during the last ten years, which has applied his methods to fundamental 

 problems in botany and zoology, has richly demonstrated the importance of these 

 methods for biology and shown the possibility of basing the science of life on 

 exact mathematical methods. The branch of mathematics that is here in the first 

 place needed is the theory of probability. For this reason Pearson was obliged, 

 in attacking the problem of evolution from a mathematical point of view, to solve 

 some important problems in this theor}', that had not to that time been sufficiently 

 dealt with. He has solved a great part of these problems. Others remained un- 

 solved or only partially solved. The object of the present investigation is to treat 

 some of these problems, which are of great importance not only to biology, but to all 

 sciences based on observations of nature. I should be glad if the results obtained 

 will contribute to further develop the line of research laid out by Pearson and his 

 school. 



Taking an arbitrary individual in the living nature — a man, an animal, a 

 plant — it will generally be found impossible to find out another individual in 

 all respects identical to the one first chosen. If the difference is great, we say that 

 tlie two objects belong to different orders, classes, species, subspecies a. s. o., but 

 it is impossible to carry the classification so far, that the differences between the 

 individuals of the same sub-class would disappear. Nevertheless there is something 

 that rightly may be named classe, species a. s. o. of individuals, though the strict 

 definition of these terms is difficult and scarcely can be made without employing 

 mathematical methods. 



Let us consider a number of individuals all belonging to the same species, 

 by which term we mean for the moment the narrowest group in the classification 



