1862.] 181 



the data were derived), limit the problem, and lead to solutions 

 which are analytically valid under conditions narrower than those 

 under which the data are possible. 



But the processes of mathematical logic enable us, without any 

 addition to the actual data, to effect the required construction of the 

 problem formally formally because the hypotheses which are re- 

 garded as ultimate and independent in that construction refer to an 

 ideal state of things. The nature of the conceptions employed, and 

 their connexion with the conceptions involved in the actual statement 

 of the problem, are discussed in the paper. It is sufficient to say 

 here that, whatever difficulty there may be in these conceptions as 

 conceptions, there is nothing arbitrary in the formal procedure of 

 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 l} 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 



the coefficients 5 X , 6 2 , . . . b M b differing in the different conditions so 

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



VOL. XII. O 



