and on the Distribution of the Fixed Stars. 523 



the probability that such a star as /3 Capricorni would nowhere 

 be found ? 



2. Such a star as /£? Capricorni having been found, what is 

 the pi'obability that the law or manner of distribution was not 

 one whose consequences we should be altogether unable to fore- 

 tell? 



The first of the above questions certainly admits of a perfectly 

 definite numerical answer. Let the value of the probability in 

 question be p. It has then generally been maintained that the 

 answer to the second question is also p, and against this view 

 Prof. Forbes justly contends. I am not sui'e that the abstract 

 of Mitchell's paper which I have consulted warrants the conclu- 

 sion that he held precisely this opinion ; but it has been a pre- 

 valent one, and to Prof. Forbes belongs the honour of having 

 first called it in question. Although the source of the fallacy is 

 not a matter of much importance, I will venture to offer an ex- 

 planation of it somewhat different from that of Prof. Forbes. 



Let us state Mr. Mitchell's problem, as we may now do, in 

 the following manner : — There is a calculated probability p in 

 favour of the truth in a particular instance of the proposition, 

 If a condition A has prevailed, a consequence B has not occurred. 

 Kequired the similar probability for the proposition. If a con- 

 sequence B has occurred, the condition A has not prevailed. 



Now the two propositions are logically connected. The one 

 is the "negative conversion" of the other; and hence if either 

 is true universally, the other is so. It seems hence to have been 

 inferred, that if there is a probabUity p in a special instance in 

 favour of the former, there is the same probability p in favour of 

 the latter. But this inference would be quite erroneous. It 

 would be an error of the same kind as to assert that whatever 

 probability there is that a stone arbitrarily selected is a mineral, 

 there is the same probability that a mineral arbitrarily selected 

 is a stone. But that these probabilities are different will be evi- 

 dent from their fractional expressions, which are — 



, Number of stones which are minerals 



Number of stones 

 Number of non-minerals which are not stones 



\ 



Number of non-minerals 

 It is true that if either of these fractions rises to 1, the other 

 does also ; but otherwise they will in general differ in value. 



Does then the problem, as above stated, admit of solution ? 

 I do not say of such solution as will throw light upon the con- 

 stitution of the heavens, but of such solution as will relieve from 

 all suspicion of inconsistency the theory of probabilities. 



In reply to this question, I shall give some account of a 

 2N 2 



