Theory of Probabilities. 36& 



u 6thly. The x } s, &c, about the interpretation of which you 

 inquire, are the probabilities of ideal events in an ideal pro- 

 blem connected by a formal relation with the real one. I 

 should fully concede that the auxiliary probabilities which are 

 employed in my method always refer to an ideal problem ; but 

 it is one, the form of which, as given by the calculus of logic, is 

 not arbitrary. Nor does its connexion with the real problem 

 appear to me arbitrary. It involves an extension, but as it 

 seems to me, a perfectly scientific extension, of the principles of 

 the ordinary theory of probabilities. On this subject, however, 

 I have but little to add to what I have said, Transactions of 

 the Royal Society of Edinburgh, vol. xxi. part 4, ' On the Ap- 

 plication of the Theory of Probabilities, &c/ 



" 7thly. The problem, as stated by me, and then modified by 

 the simple introduction of the hypothesis of the independence of 

 A and B, must admit of solution by my method ; and that solu- 

 tion ought to impose no restriction beyond the conditions of 

 possible experience noted in (M). 



" I should be extremely glad if mathematicians would examine 

 the analytical questions connected with the application of my 

 method. There can, I think, after the partial proofs which I 

 have given, exist no doubt that the conditions of applicability of 

 the solutions are always identical with the conditions of consist- 

 ency in the data, i. e. with what I have called, in the paper above 

 referred to, the conditions of possible experience. The proof of 

 the general proposition would involve the showing that a certain 

 functional determinant consists solely of positive terms, with 

 some connected theorems which appear to me to be of consider*- 

 able analytical interest. 



" 8thly. I certainly think your paper deserving of publication. 

 If you think proper to add any or the whole of my remarks, you 

 can do so, with of course any comments of your own." 



I remark upon Prof. Boole's observations : — 



1st. I do assume that the causes A and B are absolutely in- 

 dependent of, and uninfluenced by each other ; viz. not only the 

 probability of A acting, but also the probability of its acting 

 efficiently, are each of them the same whether B does not act, or 

 acts inefficiently, or acts efficiently ; and the like for B. 



2ndly. I do assume that the same experience which establishes 



Prob. A=«, Prob. B = /3, 

 would in the long run establish 



Prob. AB = «/3; 

 if it does not, cadit qucestio, the causes are not independent. 



