356 PROBLEMS ON CAUSES. [CHAP. XX. 



Whence 



M = Wi + 77Z 2 . . 4- m n , 



N = ^ + n z . . + n n , 

 M! = T/Z! , NI = W] , &c. 

 Substituting, we have 



wzj wz n HI n n ,_ 



Cn/^n ^(1-pj) <V(1-J>,) 



Hence we find 



T/I! + m z . . + m n ._ _ AT 



c 2 p 2 . . + c n p n 



or M 



= M + N. 



Hence, by (3), 



= ^p, . ,-f c n p n , 

 a known result. 



There are other particular cases in which the system (4) ad- 

 mits of ready solution. It is, however, obvious that in most 

 instances it would lead to results of great complexity. Nor does 

 it seem probable that the existence of a functional relation among 

 causes, such as is assumed in the data of the general problem, will 

 often be presented in actual experience ; if we except only the 

 particular cases above discussed. 



Had the general problem been modified by the restriction 

 that the event E cannot occur, all the causes A l . . A n being ab- 

 sent, instead of the restriction that the said causes cannot all fail, 

 the remaining condition denoted by the equation ( A l9 . . A n ) = 1 

 being retained, we should have found for the final logical equation 



2(X) being, as before, equal to (x l9 . . # n ), but ^ (XT) formed 

 by rejecting from < the particular constituent ^ . . ~x n if therein 

 contained, and then multiplying each ^-constituent of the result 

 by the corresponding ^-constituent. It is obvious that in the par- 

 ticular case in which the causes are mutually exclusive the value 

 of Prob. z hence deduced will be the same as before. 



18. PROBLEM IX. Assuming the data of any of the pre- 



