300 OF STATISTICAL CONDITIONS. [CHAP. XIX. 



Hence we have 



Minor limit of n (xyz) = n (x) + n (y) + n (z) - 2n (1). 



By extending this mode of reasoning we shall arrive at the 

 following conclusions : 



1st. The major numerical limit of the class represented by 

 any constituent will be found by prefixing n separately to each 

 factor of the constituent, and taking the least of the resulting 

 values. 



2nd. The minor limit will be found by adding all the values 

 above mentioned together, and subtracting from the result as 

 many, less one, times the value of n(l). 



Thus we should have 



Major limit ofnxy (1 - z) = least of the values nx, ny, and n(l - z). 

 Minor limit of nxy(\ - z) = n (x) + n (y) + n (1 - z) - 2n(l) 



= nx + n(y) -n(z) - n(l). 



In the use of general symbols it is perhaps better to regard all 

 the values n (#), n (y), n (1 - z), as major limits of n (xy (1 - z)} 9 

 since, in fact, it cannot exceed any of them. I shall in the fol- 

 lowing investigations adopt this mode of expression. 



PROPOSITION II. 



6. To determine the major numerical limit of a class expressed 

 by a series of constituents of the symbols #, y, z, *c., the values of 

 n(x), n(y), n(z), r., andn(l), being given. 



Evidently one mode of determining such a limit would be to 

 form the least possible sum of the major limits of the several con- 

 stituents. Thus a major limit of the expression 



would be found by adding the least of the two values nx, ny, fur- 

 nished by the first constituent, to the least of the two values 

 n (1 - #), n (1 - y), furnished by the second constituent. If we 

 do not know which is in each case the least value, we must form 

 the four possible sums, and reject any of these which are equal to 

 or exceed n (1). Thus in the above example we should have 



