352 PROBLEMS ON CAUSES. [CHAP. XX. 



probably sufficient, may also be established by other conside- 

 rations. 



Thus from the demonstration of the general method in pro- 

 babilities, Chap. XVII. Prop, iv., it appears that the quantities 

 #i> %M ^u I m the primary system of algebraic equations, 

 must be positive proper fractions. Now 



Hence generally n r must be a positive quantity, and therefore 

 we must have 



v>c r (l -p r ). 



In like manner since we have 



x r t r c r p r 



we must have generally 



fJL > C r p r . 



16. It is probable that the two classes of conditions thus re- 

 presented are together sufficient to determine generally which of 

 the roots of the equations determining ju and v are to be taken. 

 Let us take in particular the case in which n = 2. Here we have 



i / v .. 



jU + V - 1 = -- - = fJL ~ (dpi + C z p z ) + 



..v = - c l p l - c z p z + - --- - = - c^ - -, 



Whence, since juL>dp\ we have generally 



v < I - dpi- 

 In like manner we have 



v < 1 - c 2 /> 2 , p<l-d(l-pi)9 AI < 1 - c 2 (l -p z ). 



Now it has abeady been shown that there will exist but one 

 value of fj. satisfying the whole of the above conditions relative 

 to that quantity, viz. 



M > C r p r , JU < 1 - C r (1 - /?,.), 



whence the solution for this case, at least, is determinate. And I 



