340 PROBLEMS ON CAUSES. [CHAP. XX, 



we find 



Prob. z = 



and 



l + m 



-f m l 4- 



m^rii + m l _ m^n^ + m z _ , , 



whence, if we assume, 



(m, + 1) (m 2 + 1) = M, (X + 1) (n z + 1) = ^, (9) 



we have, after a slight reduction, 

 Prob.z = -r- 



Cl - /?! C z 1 - 



or 



m 2 M 



+ 1) c^ (m 2 + 1) c 2 j 2 (^ + 1) Ci (1 - 



Now let a similar series of transformations and reductions be 

 performed in thought upon the final logical equation (5). We 

 shall obtain for the determination of Prob. z the following ex- 

 pression : 



wherein 



M = (Wi + 1) (m 2 + 1) . . (m n + 1), 



N = (! + 1) (w 2 +!)..(+ 1), 



. . win, ii, . . w, being given by the system of equations, 



m n M 



(m 2 + 1) c 2 /? 2 ' (w n + l)c n p n 



00 



Still further to simplify the results, assume 



