CHAP. XX.] PROBLEMS ON CAUSES. 359 



Represent the combination #1 # 2 x n by w, then we have the 

 following logical equations : 



t(l - O = 0, t(l - O = . . t(\ - Xn ) = 0, 



#1 #2 . . X n W. 



Reducing the last to the form 



(#1 X 2 . X n ) (1 - W) + W (1 - X l # 2 X n ) = 0, 



and adding it to the former ones, we have 



S(l - Xi) + #! a? 2 . x n (1 - M?) + w (1 - a?i x z . . x n ) = 0, (1) 



wherein S extends to all values of i from 1 to w, for the one logi- 

 cal equation of the data. With this we must connect the nume- 

 rical conditions, 



Prob. a?j = Prob. # 2 = Prob. x n = p, Prob. t = a ; 

 and our object is to find Prob. w. 

 From (1) we have 





, , , , 

 ( 



on developing with respect to t. This result must further be 

 developed with respect to x l9 o; 2 , . . x n . 



Now if we make x l = 1, x 2 = 1, . . x n = 1, the coefficients both 

 of t and of 1 - 1 become 1. If we give to the same symbols any 

 other set of values formed by the interchange of and 1, it is 

 evident that the coefficient of t will become negative, while that 

 of 1 - 1 will become 0. Hence the full development (2) will be 



w = a?! x 2 . . x n t + Xi x z . . x n (1 - t) + (1 - x l x 2 . . x n ) (1 - 1) 



+ constituents whose coefficients are -, or equivalent to -. 

 Here we have 



V '= X l X^..X n t-\-X l SS 2 . .X n (l-t) + (\-X l X z ..X n ) (1 - t) 



= Xi a? 2 . . x n t + I - t ; 

 whence, passing from Logic to Algebra, 



