612 PROFESSOR BOOLE ON THE COMBINATION 



those, which contain t as a factor, &c. ; and then regarding s, t, &c, as algebraic 

 quantities. From the system thus formed, we must determine u as a function of 

 p,q . . and the arbitrary constant c, should it be present. This will be the solu- 

 tion of the problem. 



The quantities s, t . . are the same as p, q . . and represent the probabilities 

 of the hypothetical simple events, represented by s', t . . Accordingly, as pro- 

 babilities, they must admit of being determined as positive proper fractions, and 

 that the solution may not be ambiguous, they must admit of only one such de- 

 termination. These conditions will be fulfilled whensoever the problem represents 

 a possible experience, and it will be then only fulfilled. And in this way, or by 

 directly investigating the conditions of possibility by the rule of Art. 14, a solution 

 is made determinate. 



The arbitrary constant c does not, as has been intimated, always present itself. 

 When it does, it represents the unknown probability, that if the event C occur, w 

 will occur. It indicates, therefore, the new experience which would be necessary 

 in order to make the solution definite. 



18. I will, for the sake of illustration, apply the method to the problem of 

 Art. 11, and in so doing I will limit the solution by the conditions relative to 

 s, t, &c. 



The problem, as symbolically expressed in Art. 13, is as follows : — 

 Given Prob. xy=p Vvoh.yz=q Prob. zx=r ,,. 



Required Prob. xyz 

 Translating the problem as directed in the first part of the rule, we write 



wy = s yz=t zx=v j 2 



xyz—w J 



whence, by the calculus of logic, 



w = stv + Q (s t V + t S V+V S t + S V t) 



+ Q(stv + svt+tvs) . . . . (3) 



Hence we find 



Y=stv + s t v + t s v + v s t + s t v ..... (4) 



and are led to the algebraic system of equations 



stv + s tv _ stv + t s v stv + v st 



p q r 



stv __ -. 



= ^~ — stv + s tv + tsv + vst + stv . • • \°) 



These equations may be simplified by dividing every term by s 1 v, and then as- 

 suming 



JL =S > l =t ^=v' . . . . (6) 



stv 

 They thus give 



s'tv' + s' s't'v' + t _ s'tfv' + v' 



p q r 



'/./ 



= il^l = s ' t v + s' + f' + v'+l . (7) 



u 



