OF LOGICAL CLASS-FREQUENCIES, ETC. 
133 
Now ifpj + ps > 1 the lower limit to z is, by the inferior conditions of consistence, 
+ —1. But (1) and (2) both give lower limits still, and therefore do not come 
into account. Again, if pg -|- < 1, the lower limit to z, by the inferior conditions, 
is zero. But (1) and (2) give negative limits, and therefore again do not come into 
account. 
As regards the major limits, z must not, by the inferior conditions, be greater than 
the least of V-i and But if7:>2 1®®® (3) and (4) both give limits greater than 
^ 2 ; if Pi be the less (3) and (4) both give limits greater than _233. In neither case 
then do they come into account. We have therefore proved the theorem. 
