330 PROBLEMS ON CAUSES. [CHAP. XX. 



Similarly 



bq a (1 - p) , b(l - q) 



V' n= y n = _\(i y 



Hence JJL v s 



m + 1 = - , n + I = r, &c. 



ILL- ap v-a(l-p) 



Substitute these values in (2) reduced to the form 



fJ. V 



+ H-1 



+l) (n+ 1) (v> 1)' 

 and we have 



i _- 



" jU V 



Substitute also for m, TTZ', &c. their values in (3), and we have 



Prob. xy 



"I , , 



- J ^ + ^ 



v- 



b (4)> 



Now the first equation of the system (4) gives 



ap 



V 



abpq 



.'. r** = v - 1 + ap + bq. 



Similarly, 



a /K\ 



(5) 



Adding these equations together, and observing that the first 

 member of the result becomes identical with the expression just 

 found for Prob. xy> we have 



Prob. xy = v + n + a + b- e l. 

 Let us represent Prob. xy by u, and let a + b - 2 = m, then 



ju + v = u - m. (6) 



Again, from (5) we have 



.fj.v = abpq - (ap + bq - 1) ^. (7) 



