CHAP. XIX.] OF STATISTICAL CONDITIONS. 313 



Hence E' will be formed from the constituents in 1 - 0, i. e. 

 from the constituents not found in in the same way as E is 

 formed from the constituents found in 0. 



Consider next the equation 



Ew + E'(\ -w) = 0. 

 This gives 



' ' 



Now -Z? and ." are functions of the symbols Si , * 2 . . s n . The 

 expansion of the value of w will, therefore, consist of all the con- 

 stituents which can be formed out of those symbols, with their 

 proper coefficients annexed to them, as determined by the rule 

 of development. 



Moreover, E and E' are each formed by the multiplication of 

 factors, and neither of them^can vanish unless some one of the 

 factors of which it is composed vanishes. Again, any factor, as 

 ~s : . . + s n can only vanish when all the terms by the addition of 

 which it is formed vanish together, since in development we at- 

 tribute to these terms the values and 1, only. It is further evi- 

 dent, that no two factors differing from each other can vanish 

 together. Thus the factors Hi 4- s 3 . . + H n , and $i + ~s z . . + ? , can- 

 not simultaneously vanish, for the former cannot vanish unless 

 s 1 = 0, or $i = 1 ; but the latter cannot vanish unless s x = 0. 



First, let us determine the coefficient of the constituent 

 "siS 2 *n in the development of the value of w. 



The simultaneous assumption Jj = 1, ? 2 = 1 . . ? = 1, would 

 cause the factor s t + s z . . + s n to vanish if this should occur in 

 E or E'-, and no other factor under the same assumption would 

 vanish ; but Si + s z . . + s n does not occur as a factor of either 

 E or jE 7 ; neither of these quantities, therefore, can vanish; and, 



TfT ft 



therefore, the expression -- =,, is neither 1,0, nor -. 

 L - XL 



Wherefore the coefficient of J x <F 2 . . ~s n in the expanded value 

 of w, may be represented by - . 



Secondly, let us determine the coefficient of the constituent 



