CHAP. XIX.] OF STATISTICAL CONDITIONS. 297 



And generally any such statistical relations as the above will be 

 converted into relations connecting the probabilities of the events 

 concerned, by changing n(l) into 1, and any other symbol n(x) 

 into Prob. x. 



3. First, then, we shall investigate a method of determining 

 the numerical relations of classes or events, and more particularly 

 the major and minor limits of numerical value. Secondly, we 

 shall apply the method to the limitation of the solutions of ques- 

 tions in the theory of probabilities. 



It is evident that the symbol n is distributive in its operation. 

 Thus we have 



n(ay+(l-x) (l-y)} = nxy + n(\ -x) (1 -y) 

 nx (1 - y) = nx - nxy, 



and so on. The number of things contained in any class re- 

 solvable into distinct groups or portions is equal to the sum of 

 the numbers of things founcl in those separate portions. It is 

 evident, further, that any expression formed of the logical sym- 

 bols x, y, &c. may be developed or expanded in any way consis- 

 tent with the laws of the symbols, and the symbol n applied to 

 each term of the result, provided that any constant multiplier 

 which may appear, be placed outside the symbol n\ without affect- 

 ing the value of the result. The expression n (1), should it ap- 

 pear, will of course represent the number of individuals contained 

 in the universe. Thus, 



n (1-tf) (l-y) = n(l -x-y + xy) 

 n (1) - n (x) - n (y) + n (xy). 



Again, n [xy + (1 - a) (1 - y)} = n (1 - x - y + 2xy) 

 = n (1) - nx - ny + 2nxy). 



In the last member the term 2nxy indicates twice the number of 

 individuals contained in the class xy. 



4. We proceed now to investigate the numerical limits of 

 classes whose logical expression is given. In this inquiry the 

 following principles are of fundamental importance : 



1st. If all the members of a given class possess a certain pro- 

 perty # , the total number of individuals contained in the class x 



