CHAP. XX.] PROBLEMS ON CAUSES. 343 



Evidently aj M # 2 > #/n will be proper fractions, and we have 



X n -X l X 3 ..X n .. -#! 0? a #-! +71-0! X 2 . . X n 



- 1 - (1 - #1) x z #3 . . x n - Xi (1 - x 2 ) #3 . . x n . . 

 -XiXt.. x n . t (1 - x n ) - ^ a?., . . o? tt . 



Now fhe negative terms in the second member are (if we may 

 borrow the language of the logical developments) constituents 

 formed from the fractional quantities #1, # 2 , . . # n . Their sum 



cannot therefore exceed unity ; whence ~ is positive, and ju in- 



creases with u between the limits specified. 

 Now let (14) be written in the form 



and assume u - a l . The first member becomes 



(18) 



and this expression is negative in value. For, making the same 

 assumption in (15), we find 



(bi - u) .. (b n - u) 

 fj. - ! = * - ' ^ n _ t - '- = a positive quantity. 



At the same time we have 



(jic - a z ) . . (ju - an) 



H ~ l 



fJL fJ, fl 



and since the factors of the second member are positive fractions, 

 that member is less than unity, whence (18) is negative. Where- 

 fore the assumption u = i makes the jftrst member of (17} ne- 

 gative. 



Secondly, let u = ^ , then by (15) /j. = u = h , and the first mem- 

 ber of '(17) becomes positive. 



Lastly, between the limits u = i and u = bi, the first member 

 of (17) continuously increases. For the first term of that ex- 

 pression written under the form 



