CHAP. XX.] PROBLEMS ON CAUSES. 375 



will be the case, and that moreover the m + I th drawing will 



l\m + i 



, whence the probability, that if the 



first m drawings yield white balls only, the m + I th drawing will 

 also yield a white ball, is 



and generally, any proposed result will have the same probability 

 as if' it were an even chance whether each particular drawing 

 yielded a white or a black ball. This agrees with the conclusion 

 before obtained. 



26. These results only illustrate the fact, that when the defect 

 of data is supplied by hypothesis, the solutions will, in general, 

 vary with the nature of the hypotheses assumed ; so that the 

 question still remains, only more definite in form, whether the 

 principles of the theory of probabilities serve to guide us in the 

 election of such hypotheses. I have already expressed my convic- 

 tion that they do not a conviction strengthened by other reasons 

 than those above stated. Thus, a definite solution of a problem 

 having been found by the method of this work, an equally de- 

 finite solution is sometimes attainable by the same method when 

 one of the data, suppose Prob. x = p l is omitted. But I have not 

 been able to discover any mode of deducing the second solution 

 from the first by integration, with respect to p supposed variable 

 within limits determined by Chap. xix. This deduction would, 

 however, I conceive, be possible, were the principle adverted to 

 in Art. 23 valid. Still it is with diffidence that I express my 

 dissent on these points from mathematicians generally, and more 

 especially from one who, of English writers, has most fully en- 

 tered into the spirit and the methods of Laplace ; and I venture 

 to hope, that a question, second to none other in the Theory of 

 Probabilities in importance, will receive the careful attention 

 which it deserves. 



