CHAP. XIX.] OF STATISTICAL CONDITIONS. 301 



nx + n(\ -x) = w(l). 



n(x) + n(\ -y) = n(\) + n(x) - n(y). 



n(y) +n(l-y) = n(l) + w(y) - rc(#). 



Rejecting the first and last of the above values, we have 

 n (1) -f n (x) - n (y), and n (1) + n (y) - n (x), 

 for the expressions required, one of which will (unless nx = ny) 

 be less than n(l), and the other greater. The least must of 

 course be taken. 



When two or more of the constituents possess a common fac- 

 tor, as x, that factor can only, as is obvious from Principle I., 

 furnish a single term n (x) in the final expression of the major 

 limit. Thus if n (x) appear as a major limit in two or more con- 

 stituents, we must, in adding those limits together, replace 

 nx + nx by nx, and so on. -Take, for example, the expression 

 n {xy + x (1 - y)z}. The major limits of this expression, imme- 

 diately furnished by addition, would be 



1. nx. 4. ny + nx. 



2. nx + n (1 - y). 5. ny + n (1 - y). 



3. nx + n (z). 6. ny + nz. 



Of these the first and sixth only need be retained ; the second, 



third, and fourth being greater than the first ; and the fifth being 



equal to n (1). The limits are therefore 



n (x) and n (y} + n (z), 



and of these two values the last, supposing it to be less than n (1), 



must be taken. 



These considerations lead us to the following Rule : 



RULE. Take one factor from each constituent, and prefix to 



it the symbol w, add the several terms or results thus formed toge- 



ther, rejecting all repetitions of the same term ; the sum thus ob- 



tained will be a major limit of the expression, and the least of all 



such sums ivill be the major limit to be employed. 

 Thus the major limits of the expression 



xyz + *(1 -y) (1 - z) + (1 - x) (l-g) (1 - z) 



would be 



