Mr. G. Boole on the Theory of Probabilities. 315 



thought with which they are connected. The probabilities of the 

 ideal events enter as auxiliary quantities into the process of solution, 

 and disappear by elimination from the final result, but they are 

 throughout treated as probabilities, and combined according to the 

 laws of probabilities. I will only say here that the difficulty which 

 has been felt in the conception of the ideal events appears to me to 

 arise from a misdirected attempt to conceive those events by means 

 of the events in the statement of the problem — the true order of 

 thought being that the events in the statement of the problem are, 

 not indeed in their material character, but as subjects of probability 

 and of relations affecting probability, to be conceived by means of 

 the ideal events. 



Now the probabilities which constitute the actual data will in gene- 

 ral be subject to conditions in order that they may be derived from 

 actual experience. Those conditions admit of mathematical expres- 

 sion. 



Generally, if the events in the data are all or any of them compound, 

 and if p v p 2 , . . . p n represent their probabilities, those quantities will 

 be subject to certain conditions, expressible in the form of linear 

 equations or inequations, beside the condition that, as representing 

 probabilities, they must be positive proper fractions. All such condi- 

 tions of either kind are ultimately expressible in the general form 



b iPl + b 2 p 2 ...+b nPn +b=0, 



the coefficients b x , b 2 , . . . b n > b differing in the different conditions so 

 as to indicate that each of the quantities p v p 2 , . . . p n varies between 

 a system of inferior limits expressed by linear functions of the other 

 quantities, and a system of superior limits also so expressed. 



Thus, if A, B, C represent any simple events, and if p x represent 

 the probability of the concurrence of B and C, p 2 that of the concur- 

 rence of C and A, p 3 that of the concurrence of A and B, then p v p 2 , p 3 

 must, in order that they may be derived from experience, satisfy the 

 conditions 



Pl>P2+Ps— l > P^Pz+Pl- l > P^Pi+P*- 1 * 



as well as the conditions implied in their being positive proper frac- 

 tions. 



On the other hand, the ideal events being by hypothesis simple and 

 independent, the auxiliary quantities which represent their proba- 

 bilities will be subject to no other condition a priori than that of 

 being positive proper fractions — to no other condition a 2^'io?i, 

 because their actual values are determined in the process of solu- 

 tion. 



Now the most general results of the analytical investigation are — 



1st. That the auxiliary quantities representing the probabilities of 

 the ideal events admit of determination as positive proper fractions, 

 and, further, of a single definite determination as such, precisely when 

 the original data supply the conditions of a possible experience. 



2ndly. That as a consequence of this the probability sought will 



