82 
ON A QUESTION IN THE THEORY OF PROBABILITIES. 
[318 
4thly. You remark that my solution of the problem, in which the independence 
of A and B is not assumed, but in which the probabilities are otherwise the same 
as in yours, is only applicable when 
+ a P !> fiq, /3' + ¡3q 5 ap ; 
but you do not appear to have noticed that these are actually the conditions of 
consistency in the data. Unless these are satisfied, the data cannot possibly be furnished 
by experience. 
othly. You remark that I have solved the problem under what you call the 
‘ concomitance ’ statement, and not the ‘ causation ’ statement. I think that every problem 
stated in the ‘ causation ’ form admits, if capable of scientific treatment, of reduction 
to the ‘ concomitance ’ form. I admit it would have been better, in stating my problem, 
not to have employed the word ‘ cause ’ at all. But the introduction of the hypothesis 
of the independence of A and B does not affect the nature of the problem. 
Gthly. The x, s, &c., about the interpretation of which you inquire, are the pro 
babilities of ideal events in an ideal problem 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 pro 
babilities. 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 Application 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 solution 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 consistency 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 considerable 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 independent of, and 
uninfluenced by each other; viz. not only the probability of A acting, but also the