CHAP. XX.] PROBLEMS ON CAUSES. 325 



Between those limits V varies continuously without becoming 



d z V 

 infinite, and -=-^ is always of the same sign. 



Hence if u represent the abscissa V the ordinate of a plane 

 curve, it is evident that the curve will pass from a point below 

 the axis of u corresponding to u = a, to a point above the axis of 

 u corresponding to u = ', the curve remaining continuous, and 

 having its concavity or convexity always turned in the same di- 

 rection. A little attention will show that, under these circum- 

 stances, it must cut the axis of u once, and only once. 



Hence between the limits u = , u = ', there exists one value 

 ofw, and only one, which satisfies the equation (11). It will 

 further appear, if in thought the curve be traced, that the other 

 value of u will be less than a when the quantity 1 - a - b' is po- 

 sitive and greater than any one of the quantities a', #, c' when 

 1 - a - b' is negative. It hence follows that in the solution of 

 (11) the positive sign of the radical must be taken. We thus 

 find 



ab-a'b'+(l-a'-b')c+)/Q , . 



2(1 - a -b') 



where Q= {ab-a'b'+(l -a'-6>'} 2 - 4(1 -a-b')(ab-ab'c). 



4. The results of this investigation may to some extent be 

 verified. Thus, it is evident that the probability of the event E 

 must in general exceed the probability of the concurrence of the 

 event E and the cause A^ or A z . Hence we must have, as the 

 solution indicates, 



u > Cii 9 u 



Again, it is clear that the probability of the effect E must in 

 general be less than it would be if the causes A lt A z were mu- 

 tually exclusive. Hence 



U < C^i + C 2 J0 2 . 



Lastly, since the probability of the failure of the effect ^con- 

 curring with the presence of the cause A 1 must, in general, be 

 less than the absolute probability of the failure of E, we have 



Ci (1 - pi) < 1 - u, 



