CHAP. XIX.] OF STATISTICAL CONDITIONS. 295 



CHAPTER XIX. 



OF STATISTICAL CONDITIONS. 



1. ~P) Y the term statistical conditions, I mean those conditions 

 J-J which must connect the numerical data of a problem in 

 order that those data may be consistent with each other, and 

 therefore such as statistical observations might actually have 

 furnished. The determination of such conditions constitutes an 

 important problem, the solution of which, to an extent sufficient 

 at least for the requirements of this work, I purpose to undertake 

 in the present chapter, regarding it partly as an independent ob- 

 ject of speculation, but partly also as a necessary supplement to 

 the theory of probabilities already in some degree exemplified. 

 The nature of the connexion between the two subjects may be 

 stated as follows : 



2. There are innumerable instances, and one of the kind 

 presented itself in the last chapter, Ex. 7, in which the solution 

 of a question in the theory of probabilities is finally dependent 

 upon the solution of an algebraic equation of an elevated degree. 

 In such cases the selection of the proper root must be determined 

 by certain conditions, partly relating to the numerical values as- 

 signed in the data, partly to the due limitation of the element 

 required. The discovery of such conditions may sometimes be 

 effected by unaided reasoning. For instance, if there is a proba- 

 bility p of the occurrence of an event A, and a probability q of 

 the concurrence of the said event -4, and another event J5, it is 

 evident that we must have 



But for the general determination of such relations, a distinct 

 method is required, and this we proceed to establish. 



As derived from actual experience, the probability of any 

 event is the result of a process of approximation. It is the limit 

 of the ratio of the number of cases in which the event is observed 

 to occur, to the whole number of equally possible cases which 



