350 PROBLEMS ON CAUSES. [CHAP. XX. 



1 _(n 



~ 



The above value of Prob. w will be the numerator of the fraction 

 (2). It now remains to determine its denominator. 

 For this purpose assume 



or = v ; 



whence 0v + t^> = 0* 



Substituting the first member of this equation in (7) in place of 

 the corresponding form <f>zw + w (1 - <f>z) we obtain as the primary 

 logical equation, 



S [0! r zJ r + t r (1 - tfrZ) ) + Xi . . X n Z + 0U + V0 = 0, 



whence eliminating 2, and reducing by Prop. n. Chap. IX., 



(j)V + V$ + S# r { S (^?r ^r + t r ~X r ) + ~X\ . . ~X n } =0. 



Hence 



and developing as before, 



+ - (sum of other constituents). (12) 



Here Si (X) indicates the sum of all constituents found in 0, 

 S 2 (-3Q the sum of all constituents not found in 0. The expres- 

 sions are indeed used in place of and 1 - to preserve sym- 

 metry. 



It follows hence that Si (X) + S 2 (X) = 1, and that, as be- 

 fore, Si (X T) + S 2 (X T) = S (X T). Hence F will have the 

 same value as before, and we shall have 



Prob ... 



Or transforming, as in the previous case, 



