CHAP. XX.] PROBLEMS ON CAUSES. 331 



Similarly from the first and third members of (4) equated we 

 have 



VLV = ab(l -p)(\-q)- (a(\-p) + b(l - q) - 1} v . 



Let us represent ap + bq - 1 by h, and a (1 - p) + b (1 - ^) - 1 by 

 ^'. We find on equating the above values of /xv, 



fyi - h'v = ab {pq + (l-p)(l-q)} 

 = ab(p + q- 1). 



Let ab (p + q - 1) = /, then 



hfi - h' v = /. (8) 



Now from (6) and (8) we get 



h' (u - m) + I h (u - m) - I 



p = - L - v = -- '- - . 



m m 



Substitute these values in (7) reduced to the form 



fj. (v + h) = abpq, 

 and we have 



(hu - /) {h' (u - m) + 1} = abpqm^ (9) 



a quadratic equation, the solution of which determines u, the va- 

 lue of Prob. xy sought. 



The solution may readily be put in the form 



IV + h(h'm -l) 



But if we further observe that 



IK - h (Km -l) = l(h + h')- hh'm = (/ - hK) m, 

 since h = ap + bq 1, K = a (1 - p) + b (1 - q) 1, 

 whence h-th'=a-i-b-2 = m ) 



we find 



P roh - ^' + KP''"-0 + m V {(l-hKy + lhKabpq] 

 ^~ 2M' ' 



It remains to determine which sign must be given to the radi- 

 cal. We might ascertain this by the general method exemplified 

 in the last problem, but it is far easier, and it fully suffices in the 

 present instance, to determine the sign by a comparison of the 



