CHAP. XIX.] OF STATISTICAL CONDITIONS. 307 



required the systems of superior and inferior limits of Prob. iv, 

 and the conditions among the data. 



SOLUTION. The superior limits of Prob. (A + (7), and the 

 inferior limits of Prob. {A + D) will form two such systems as are 

 sought. The conditions among the constants in the data will be 

 given by the inequality, 



Inf. lim. Prob. D < 0. 



In the application of these principles we have always 

 Inf. lim. Prob. x l x 2 . . x n = Prob. a?! + Prob. x 2 . . + Prob.# n - (n - 1 ). 



Moreover, the inferior limits can only be determined from single 

 terms, either given or formed by aggregation. Superior limits 

 are included in the form S Prob. x, Prob. x applying only to 

 symbols which are different, and are taken from different terms in 

 the expression whose superior limit is sought. Thus the supe- 

 rior limits of Prob. xyz + x (1 - y) (1 - z) are 



Prob. x, Prob. y + Prob. (1 - z), and Prob. z + Prob. (1 -y). 



Let it be observed, that if in the last case we had taken Prob. z 

 from the first term, and Prob. ( 1 - 2) from the second, a con- 

 nexion not forbidden, we should have had as their sum 1, which 

 as a result would be useless because a priori necessary. It is 

 obvious that we may reject any limits which do not fall between 

 and 1. 



Let us apply this method to Ex. 7, Case in. of the last 

 chapter. 



The final logical solution is 



1-1 



x = - stu -f - stu + - stu + stu 



+ -~stu + Q~siu + Q~stu + Q~stu 9 

 the data being 



Prob. s = p, Prob. t = q, Prob. u = r. 



We shall seek both the numerical limits of x, and the condi- 

 tions connecting p, q, and r. 



x2 



