CHAP. XX.] PROBLEMS ON CAUSES. 347 



S [XrZ7 r + t r (1 -X r z)} + *i . . ~X n Z + $ZW + W (1 - 0z) = 0, (7) 



from which z being eliminated, w must be determined as a de- 

 veloped logical function of x l , . . x n , *i ?. 



Now making successively z = 1, 2 = in the above equation, 

 and multiplying the results together, we have 



{ S (x r T r + !c r tr) +xi..x n + <j>w+w<j>}x (S r + w) = 0. 



Developing this equation with reference to w?, and replacing 

 in the result S r + 1 by I, in accordance with Prop. i. Chap, ix., 



we have 



Ew + E' (1 - w) = ; 

 wherein 



E = S (X r j r + t r !H; r ) + lCi . . X n + 0, 

 .#'= S* r {2 (X r j r 4- ^ r ) + ^ . . X n + $} . 



And hence 



The second member of this equation we must now develop 

 with respect to the double series of symbols a? 19 # 2 , . .x n , t l9 t z , . .t n . 

 In eifecting this object, it will be most convenient to arrange 

 the constituents of the resulting development in three distinct 

 classes, and to determine the coefficients proper to those classes 

 separately. 



First, let us consider those constituents of which Ji . . 7 n is a 

 factor. Making ti = . . t n = 0, we find 



E' = 0, E = Stf r + :Fi ..:? + 0. 



It is evident, that whatever values (0, 1) are given to the ^-sym- 

 bols, J^does not vanish. Hence the coefficients of all constituents 

 involving ?! . . ~t n are 0. 



Consider secondly, those constituents which do not involve the 

 factor ?!..?, and which are symmetrical with reference to the two 

 sets of symbols Xi . . x n and t\ . . t n . By symmetrical constituents 

 is here meant those which would remain unchanged if x l were 

 converted into t l9 # 2 into t z , &c., and vice versa. The constitu- 

 ents #1 # *i . . f> *i # *i ' %j &c., are in this sense sym- 

 metrical. 



