308 OF STATISTICAL CONDITIONS. [CHAP. XIX. 



The superior limits of x are, according to the rule, given by 

 those of stu + stu. They are, therefore, 



p 9 q + 1 - r, r + I - q. 

 The inferior limit of x are given by those of 

 stu + stu, + stu -f ~stu. 



We may collect the first and third of these constituents in the 

 single term st. and the second and third in the single term su. 

 The inferior limits of x must then be deduced separately from 

 the terms s (1 - Q, s (1 - u) 9 (1 - s) tu 9 which give 



p + I - q - I, p+l-r-l, l-p + q + r-2 9 

 or p - q 9 p - r, and q + r - p - 1 . 



Finally, the conditions among the constants p 9 q, and r, are 

 given by the terms 



stu, stUy ~stu, 



from which, by the rule, we deduce 



p+l-q+r-2<0 9 p + q+l-r-2<0 9 I-p+q + r-2<Q. 

 ml + q-p-r>Q 9 l+ r -p-q>0 9 l+ p -q- r = Q t 



These are the limiting conditions employed in the analysis of 

 the final solution. The conditions by which in that solution A is 

 limited, were determined, however, simply from the conditions 

 that the quantities s 9 t, and u should be positive. Narrower 

 limits of that quantity might, in all probability, have been de- 

 duced from the above investigation. 



12. The following application is taken from an important pro- 

 blem, the solution of which will be given in the next chapter. 

 There are given, 



Prob. x = c l9 Prob. y = c 2 , Prob. s = c, p l9 Prob. t = c 2 p z , 

 together with the logical equation 



z = stocy + sixy + Htxy + Q~st 

 1 f stxy + stxy + stxy + stxy + stxy 

 v [_ -f st~xy + ~stxy + Htxy -f "stxy ; 



