342 PROBLEMS ON CAUSES. [c HAP. XX. 



If we make for simplicity 



Cipi = ai, c n p n = a n , 1 - c, (1 -p,) = b l9 &c., 

 the above equations may be written as follows : 



^ M >-<>^-<>, (14) 



wherein 



This value of p substituted in (14) will give an equation in- 

 volving only u, the solution of which will determine Prob. z, 

 since by ( 13) Prob. z = u. It remains to assign the limits of u. 



1 1 . Now the very same analysis by which the limits were deter- 

 mined in the particular case in which n = 2, (XIX. 12) con- 

 ducts us in the present case to the following result. The quan- 

 tity U, in order that it may represent the value of Prob. z, must 

 must have for its inferior limits the quantities a l9 2 , . . a n9 and 

 for its superior limits the quantities 5 n > 2 ? -b n9 a \ + #2 + n 

 We may hence infer, a priori, that there will always exist one 

 root, and only one root, of the equation (14) satisfying these 

 conditions. I deem it sufficient, for practical verification, to show 

 that there will exist one, and only one, root of the equation (14), 

 between the limits a 19 25 > and b l9 b 2 , . ,b n . 



First, let us consider the nature of the changes to which ju is 

 subject in (15), as u varies from a ly which we will suppose the 

 greatest of its minor limits, to bi , which we will suppose the least 

 of its major limits. When u = a i9 it is evident that JUL is positive 

 and greater than a^ . When u = b l9 we have jj. = 6,. , which is also 

 positive. Between the limits u- a l9 u = b l9 it may be shown 

 that fjL increases with u. Thus we have 



flfri (b,-u)..(b n -u) (ft 1 - 



du l-u n ~ l 



Now let 



bi-u b-u 



