CHAP. XVII.] GENERAL METHOD IN PROBABILITIES. 267 



&c. It is obvious, therefore, that instead of the letters p' 9 q' 9 &c., 

 we might employ any others as s, t, &c., in the same quantitative 

 acceptations. This particular step would simply involve a change 

 of meaning of the symbols s, t, &c. their ceasing to be logical, 

 and becoming quantitative. The systems (7) and (8) would then 

 become 



Prob. w = -. (10) 



In employing these, it is only necessary to determine from (9) 

 s, t, &c., regarded as quantitative symbols, in terms of/?, q, &c., 

 and substitute the resulting values in (10). It is evident, that 

 s, t, &c., inasmuch as they represent probabilities, will be positive 

 proper fractions. 



The system (9) may be more symmetrically expressed in the 

 form 



P 9. 

 Or we may express both (9) and (10) together in the symme- 



trical system 



K = F, = C = 



p q u 



wherein u represents Prob. w. 



15. It remains to interpret the constant c assumed to repre- 

 sent the probability of the indefinite event q. Now the logical 

 equation 



w = A + qC, 



interpreted in the reverse order, implies that if either the event 

 A take place, or the event C in connexion with the event q, the 

 event w will take place, and not otherwise. Hence q represents 

 that condition under which, if the event C take place, the event 

 w will take place. But the probability of q is c. Hence, there- 

 fore, c = probability that if the event C take place the event w 

 will take place. 



Wherefore by Principle n., 



_ Probability of concurrence of C and w 

 Probability of C 



