CHAP. XX.] PROBLEMS ON CAUSES. 337 



Now let us assume generally 



which is reducible to the form 



forming the type of a system of n equations which, together with 

 (1), express the logical conditions of the problem. Adding all 

 these equations together, as after the previous reduction we are 

 permitted to do, we have 



2{x i z(l-t i )+t i (l-x i z)}+z(l-x l )(l-x t )..(l-x n ) = 0, (2) 



(the summation implied by S extending from i = 1 to i = ri) 9 and 

 this single and sufficient logical equation, together with the 2 n 

 data, represented by the general equations 



Prob. X{ = Ci 9 Prob. ti = c t -jt?;, (3) 



constitute the elements from which we are to determine Prob. z. 

 Let (2) be developed with respect to z. We have 



[S{*i(l - ti) + ti(l- Xi )} + (1 - O (1 - O . . (1 -O] z 



+ S*,(l-z) = 0, 



whence 



= _ Zti _ ( 

 ~ S*<- 2 lx i (l-t i ) + t i (l-x i )}-(l-x l )(l-x t )..(l^-x u y { 



Now any constituent in the expansion of the second member of 

 the above equation will consist of 2w factors, of which n are taken 

 out of the set x l9 # 2) x m 1 - x l9 1 - # 2 , . . 1 - x and n out of 

 the set t i9 Z 2 > > 1 - tiy 1 - ^> 1 - n> no such combination as 

 x l (1 - ^i), #! (1 - ti), being admissible. Let us consider first 

 those constituents of which (1 - ^), (1 - t 2 ) . . (1 - t n ) forms the 

 ^-factor, that is the factor derived from the set t l9 . . 1 - ti. 



The coefficient of any such constituent will be found by 

 changing 1? t z , . . t n respectively into in the second member of 

 (4), and then assigning to x l9 x$ 9 . . x n their values as dependent 

 upon the nature of the ^-factor of the constituent. Now simply 

 substituting for t l9 t 29 . . t n the value 0, the second member be- 

 comes 







