348 PROBLEMS ON CAUSES. [CHAP. XX. 



For all symmetrical constituents it is evident that 



/ S (X r ~t r + t r ~X r ) 



vanishes. For those which do not involve T } . . 7 n , it is further 

 evident that "xi . . ~x n also vanishes, whence 



w 



For those constituents of which the ^-factor is found in the 

 second member of the above equation becomes 1 ; for those of 

 which the ^-factor is found in it becomes 0. Hence the coeffi- 

 cients of symmetrical constituents not involving J } . . ?, of which 

 the x-f actor is found in <f> will be 1 ; of those of which the x-factor 

 is not found in it will be 0. 



Consider lastly, those constituents which are unsymmetrical 

 with reference to the two sets of symbols, and which at the same 

 time do not involve ?i . . ~t n . 



Here it is evident, that neither E nor E' can vanish, whence 

 the numerator of the fractional value of w in (8) must exceed 

 the denominator. That value cannot therefore be represented 



by 1, 0, or -. It must then, in the logical development,be re- 

 presented by - . Such then will be the coefficient of this class 



of constituents. 



15. Hence the final logical equation by which w is expressed 

 as a developed logical function of ar l5 . . x n , ti 9 . . t n , will be of 

 the form 



w = 2! (XT) + { 2 Z (XT) + T, ..!}+ i (sum of other con- 

 stituents), 



wherein Si (XT) represents the sum of all symmetrical consti- 

 tuents of which the factor X is found in 0, and 2 2 (-X^), the 

 sum of all symmetrical constituents of which the factor X is not 

 found in 0, the constituent xi . . ~x n ~ti . . T n9 should it appear, 

 being in either case rejected. 



Passing from Logic to Algebra, it may be observed, that 



