314 OF STATISTICAL CONDITIONS. [CHAP. XIX. 



The assumptions $1 = 1, s 2 = 1, . . s n - 1, would cause the factor 

 $! + s 2 - + ~$n to vanish. Now this factor is found in E and not 

 in E whenever contains both the constituents x l x 2 . . x n and 



77" 77" 



Xi x z - . x n - Here then - ^ becomes - or 1. The factor 



?i + ? 2 + sn is found in ./' and not in JS', if contains neither 

 of the constituents Xi x z . . x n and Xix z ..x n - Here then 



V 



- - -becomes ^ or 0. Lastly, the factor ^ + ? 2 . . + ? is 

 jL Hj JL/ 



contained in both ^J and E', if one of the constituents x l x z . . x n 



E' 



and xiXz. . x n is found in rf>, and one is not. Here then r - 



becomes - . 



The coefficient of the constituent s s z . . s n , will therefore be 

 1,0, or - , according as contains both the constituents x l x 2 . . x n 



and XiXz'.Xn^ or neither of them, or one of them and not the 

 other. 



Lastly, to determine the coefficient of any other constituent 

 as s l . . S L sui . .?. 



The assumptions Si = 1, . . $i = 1, s i+1 = 0, s n = 0, would 

 cause the factor 1 : . . + ^ + s i+l . . + s n to vanish. Now this fac- 

 tor is found in E, if the constituent x^ . . x- t x i+l . . x n is found in 

 <p and in E', if the said constituent is not found in 0. In the 



77" 77" 



former case we have -=$ - ^ = ^r = 1 ; in the latter case we have 

 Jbj Jit L 



E' _ Q 



E 1 - E Q-E~ 



Hence the coefficient of any other constituent ^ . . s i9 H i+l . . ?, 

 is 1 or according as the similar constituent x l . . xi xi +l . . x n 

 is or is not found in (ft. 



We may, therefore, practically determine the value of w in 

 the following manner. Rejecting from the given expression of 

 the constituents x l x z . . x n and and x l x 2 . . x n , should both or 

 either of them be contained in it, let the symbols x l , x. 2 , . . x n , 

 in the result be changed into s 19 5 2 , . .s n respectively. Let the co- 

 efficients of the constituents Sj, s 2 . . s n and sj ? 3 . . ~s n be determined 



