CHAP. XX.] PROBLEMS ON CAUSES. 329 



To reduce the above system to a more convenient form, let every 

 member be divided by Icy ~s~t, and in the result let 



xs yt x y 



=. = m, !r=. = m, = n, ^ = n. 



xs yt x y 



We then find 



mm + m + nri + n mm + m + nri + ri 

 ~a~ ~b~ 



mm + m mm + m 



ap bq 



= mm! + m + m + nri + n + ri + 1 . 

 Also, 



^ . mm! + nri 



.rrob. xy = ; -, -, ; r. 



mm + m + m + nn + n + n + 1 



These equations may be reduced to the form 



mm + m mm + In nri + n nri + ri 

 ap bq a(l-p) b(\-q) 



mm' + nri 



p 



)b * 



(m + 1) (m' + 1) + (+ 1) (w'+ 1) - 1* 

 Now assume 



.__, __. 



m 

 Then since /rm + m = 



-, - T^~7 - T^J 

 m + 1 (m + 1) (v + /x - 1) 



and so on for the other numerators of the system, we find, on 

 substituting and multiplying each member of the system by 

 v + fi - 1, the following results : 



m/m m'/uL nv riv 



~ 



(m+ 1) ap (m 



Prob. xy = (mm + nri) (v + p - 1). (3) 



From the above system we have 



m 



an ap 



, whence m = - . 



m + I fj. fj. - ap 



