CHAP. XX.] PROBLEMS ON CAUSES. 323 



From this result we find (XVII. 17), 



V '= stxy + stxy -i- Jtxy + Jtxy + Jtxy 



+ Jtxy -f stxy 

 = stxy + stxy +Jtxy + Jt. 



Whence, passing from Logic to Algebra, we have the following 

 system of equations, u standing for the probability sought : 



stxy + stxy + Jtx stxy + Jtxy + Jty 



Ci C z 



_ stxy + stxy _ stxy +Jtxy /y\ 



Cipi c z p z 



_ stxy + stxy + Htxy stxy + stxy + Jtxy + ~st _ v 



~^r ~T 



from which we must eliminate s 9 t> x, y, and V. 

 Now if we have any series'of equal fractions, as 



we know that 



la + mb + nc 



= x. 



la+ mb' + nc' 

 And thus from the above system of equations we may deduce 



Jtxy stxy Ht v 



u - c l p l u - c 2 p z I - u 



whence we have, on equating the product of the three first mem- 

 bers to the cube of the last, 



(u - dpi) (u - c 2 p 2 ) (1 - u) 

 Again, from the system (7) we have 



stx ~sty stxy 



I u Ci + c^pi \ uc> t j f c z p 2 c l p l 



whence proceeding as before 



- w) (1 - c 2 + c z p z - u) (c,p, -I- c 2 p 2 - u) 



Y 2 



= F3. (9) 



