CHAP. XX.] PROBLEMS ON CAUSES. 327 



z = stxy + - stxy + - stxy + ^sfxy 



- ~stxy + stxy + -'stxy 



Q~s~txy 

 and the final solution is 



the quantity u being determined by the solution of the equation 



(u-d) (u-b) _(a'-u) (b'-u) m 



a + b-u ~ u-a-b' + l' 



wherein a = c^p l9 b = c. 2 p 2 , a = 1 - c x (1 - />i), b' = 1 - c 2 (1 - Pz)- 



The conditions of limitation are the following : That value 

 of u must be chosen which exceeds each of the three quantities 



, ft, and a + b' - 1 , 



and which at the same time falls short of each of the three quan- 

 tities 



', b', and a + b. 



Exactly as in the solution of the previous problem, it may be 

 shown that the quadratic equation (1) will have one root, and 

 only one root, satisfying these conditions. The conditions them- 

 selves were deduced by the same rule as before, excepting that 

 the minor limit a' + b' - I was found by seeking the major limit 

 of 1 - z. 



It may be added that the constants in the data, beside satis- 

 fying the conditions implied above, viz., that the quantities a', b\ 

 and a + ft, must individually exceed , ft, and a + ft' - 1, must 

 also satisfy the condition c t + c 2 > 1 . This also appears from the 

 application of the rule. 



6. PROBLEM III. The probabilities of two events A and B 

 are a and ft respectively, the probability that if the event A take 

 place an event E will accompany it is p, and the probability that 



