324 PROBLEMS ON CAUSES. [CHAP. XX. 



Equating the values of F 3 in (8) and (9), we have 



= j 1 - cj. (1 -pj -u} { 1 - c 2 (1 -p z )-u} 

 which may be more conveniently written in the form 

 (u-dp^u-dfr) (l-Ci(l -pi)-u] [l-c 2 (l-p 2 )-u 



c\p\ + c 2 p z -u I -u 





From this equation the value of u may be found. It remains 

 only to determine which of the roots must be taken for this pur- 

 pose. 



3. It has been shown (XIX. 12) that the quantity u, in 

 order that it may represent the probability required in the above 

 case, must exceed each of the quantities CIJPI, c z p Z9 and fall 

 short of each of the quantities 1-^(1 - ji), 1 - c 2 (1 - jt? 2 ), and 

 c l p l + c 2 /? 2 ; the condition among the constants, moreover, being 

 that the three last quantities must individually exceed each of 

 the two former ones. Now I shall show that these conditions 

 being satisfied, the final equation (10) has but one root which 

 falls within the limits assigned. That root will therefore be the 

 required value of u. 



Let us represent the lower limits c l p-^ c z p 29 by , b respec- 

 tively, and the upper limits 1 -^(1 -pi), 1 - c a (l - p z ), and 

 Cipi + c z p z , by ', b r , c' respectively. Then the general equation 

 may be expressed in the form 



(u -a) (u- b) (1 - ?/> - (a 1 - u) (b - u) (c - u) --= 0, (11) 

 or (1 - a - b') u 2 - [ab - ab' + (1 - a - b) c} u + ab - ab'c' = 0. 



Representing the first member of the above equation by F, we 

 have 



|- * fl -'-') 02) 



Now let us suppose a the highest of the lower limits of u, a the 

 lowest of its higher limits, and trace the progress of the values 

 of V between the limits u = a and u = a. 



When u = , we see from the form of the first member of (1 1) 

 that V is negative, and when u = a we see that V is positive. 



