346 PROBLEMS ON CAUSES. [CHAP. XX. 



practical difficulty will not be great. For the conditions relating 

 to the limits enable us to select at once a near value of u, and 

 the forms of the system (19) are suitable for the processes of suc- 

 cessive approximation. 



14. PROBLEM 7. The data being the same as in the last pro- 

 blem, required the probability, that if any definite and given 

 combination of the causes A 19 A Z9 . .A n , present itself, the event 

 E will be realized. 



The cases AI, A 29 . . A n , being represented as before by 

 # 15 # 2 j n respectively, let the definite combination of them, 

 referred to in the statement of the problem, be represented by 

 the (j) (x l , #2 . . x n ) so that the actual occurrence of that combi- 

 nation will be expressed by the logical equation, 



The data are 



Prob. x l = c 1? . . Prob. x n = c n9 

 Prob. a&iZ - <?!/?!, Prob. x n z = c n p n ; 



and the object of investigation is 



Prob. ft (#1, a? 2 . .~Xn) z . . 



Prob. ft (#i,# 2 x n ) 



We shall first seek the value of the numerator. 

 Let us assume, 



x l z = t l . . x n z = t n , (3) 



0(^,^2- . #)* = tl7. (4) 



Or, if for simplicity, we represent $ (x l9 x 2 #n) by 0, the last 



equation will be 



0z = w, (5) 



to which must be added the equation 



*!*,..** = <). (6) 



Now any equation x r z = t r of the system (3) may be reduced 



to the form 



x r zJ r + t r (1 - x r z) = 0. 



Similarly reducing (5), and adding the different results together, 

 we obtain the logical equation 



