CHAP. XIX.] OF STATISTICAL CONDITIONS. 315 



according to the special rules for those cases given above, and let 

 every other constituent have for its coefficient 0. The result 

 will be the value of w as a function of *i, $ 2 , . . s n . 



As a particular case, let = x^ It is required from the 

 given data to determine the probability of the event x . 



The symbol x l9 expanded in terms of the entire series of sym- 

 bols a?!, #2, . . x n9 will generate all the constituents of those 

 symbols which have x l as a factor. Among those constituents 

 will be found the constituent x l x 2 . . x n9 but not the constituent 



Hence in the expanded value of x l as a function of the sym- 

 bols Si , s 29 . . s n , the constituent Si s z . . s n will have the coefficient 



- , and the constituent 7i J 2 the coefficient - . 



If from Xi we reject the constituent x l x z . . #, the result 

 will be x l - XiX z . . x n9 and changing therein Xi into s l9 &c., we 

 have $i Si s z . . s n for the corresponding portion of the expres- 

 sion of x\ as a function of s l9 s 2 , s n . 



Hence the final expression for x is 



1__ _ 



.Sn + QS l S,..S n + -S l S z .. Sn (?) 



+ constituents whose coefficients are 0. 

 The sum of all the constituents in the above expansion whose 

 coefficients are either 1, 0, or , will be 1 - ~si~s z s- 



We shall, therefore, have the following algebraic system for 

 the determination of Prob. x l9 viz. : 



T> i Si ~ SiS 2 . . S n + CS 1 S 2 . . S n 



Prob. x l = - - =^z - - - , (8) 



1 - Si s 2 . . s n 



with the relations 



1 fz S n 



Pi ~ P* ' ~ Pn (9) 



= 1 Si S 2 . . S n = X. 



It will be seen, that the relations for the determination of 

 Si s z . . s n are quite independent of the form of the function 0, 

 and the values of these quantities, determined once, will serve 



