CHAP. XIX.] OF STATISTICAL CONDITIONS. 309 



and it is required to determine the conditions among the constants 

 15 <? 2 , /?i, j9 2 , and the major and minor limits of z. 



First let us seek the conditions among the constants. Con- 



fining our attention to the terms whose coefficients are - , we 



readily form, by the aggregation of constituents, the following 

 terms, viz. : 



s(l-x) 9 *(l-y) sq(\-t), to(l-); 



nor can we form any other terms which are not included under 

 these. Hence the conditions among the constants are, 



n(t)+n(l-y)-n (1) ^ 0, 



n (s) + n (y) + n (I - t) - 2n (1) ^ 0, 



n(t)+n (x\ + n(l-s) - 2n (1) < 0. 



Now replace n (x) by c l9 n (y) by c 2 , n (s) by dp l9 n (t) by 

 c 2 j9 2 , and n(l) by 1, and we have, after slight reductions, 



C 2 , 



Such are, then, the requisite conditions among the constants. 



Again, the major limits of z are identical with those of the 

 expression 



stxy + s(\-i)x(\-y)-t(\-s)t(\- x) y\ 



which, if we bear in mind the conditions 



n(s)<n (a?), n(i)<n (y), 

 above determined, will be found to be 



n (s) + n (t), or, c^ -f c 2 /? 2 , 

 n(s) + n(l- x), or, 1 - d (1 - /? t ). 



Lastly, to ascertain the minor limits of z, we readily form 

 from the constituents, whose coefficients are 1 or -, the single 

 terms s and t, nor can any other terms not included under these be 



