338 PROBLEMS ON CAUSES. [CHAP. XX. 



and this vanishes whatever values, 0, 1, we subsequently assign 

 to Xi, #25 #n For if those values are not all equal to 0, the 

 term 2#; does not vanish, and if they are all equal to 0, the term 

 - (1 - Oa) . . (1 - #) becomes - 1, so that in either case the denomi- 

 nator does not vanish, and therefore the fraction does. Hence 

 the coefficients of all constituents of which (1 - x ) . . (1 - n ) is a 

 factor will be 0, and as the sum of all possible ^-constituents is 

 unity, there will be an aggregate term (1 - ti) . . (1 - ) in the 

 development of z. 



Consider, in the next place, any constituent of which the 

 ^-factor is ti t z . . t r (1 - t M ) . . (1 - 4)5 r being equal to or greater 

 than unity. Making in the second member of (4), t l = 1, . . t r = 1, 

 rn - 0, . . = 0, we get the expression 



r 



Xi..+X r - X r+ i .. -X n -(l- X,) (I - X t ) . . (1 - X n )' 



Now the only admissible values of the symbols being and 1 , 

 it is evident that the above expression will be equal to 1 when 

 Xi = 1 . . x r = 1, x r+ i = 0, . . x n = 0, and that for all other combi- 

 nations of value that expression will assume a value greater than 

 unity. Hence the coefficient 1 will be applied to all constituents 

 of the final development which are of the form 



X l . . X r (1 - X r+ i) ..(!-#)*!.. t r (1 - t r+l ) . . (1 - t n ), 



the ^-factor being similar to the ^-factor, while other consti- 

 tuents included under the present case will have the virtual co- 



efficient -. Also, it is manifest that this reasoning is independent 



of the particular arrangement and succession of the individual 

 symbols. 



Hence the complete expansion of z will be of the form 



2 = S (XT) + (1 - ti) (1 - * 8 ) . . (1 - t n ) 



+ constituents whose coefficients are -, (5) 



where T represents any ^-constituent except (1 - ^) . . (1 - ), 

 and X the corresponding or similar constituent of x l . . x n . 



