90 Prof. Boole on the Theory of Chances. 



which the Solutions of Questions in the Theory of Probabilities 

 are limited," which I forward for publication with this letter. 

 On the other hand, there are no cases whatever in which the 

 problem is solvable by other methods, which do not furnish a 

 verification to the solution I have given. Now I cannot but 

 think that a cautious inquirer after truth, seeing that two hypo- 

 theses (still adopting Mr. Wilbraham's language), one of which 

 appears to him " eminently anomalous," conduct to a solution 

 which cannot by any known test be proved erroneous, while two 

 other hypotheses, which appear to him "perhaps not unreason- 

 able " (for this, Mr. Wilbraham's language already quoted implies 

 with reference to Mr. Cayley's hypotheses), conduct to a solution 

 which will not bear the test of examination, would be led to sus- 

 pect that he had been judging of the reasonableness and of the 

 anomalous character of hypotheses by some false standard. Of 

 course if a solution is eiToneous, it need not to be argued that 

 there must be error in the hypotheses by which it was obtained. 

 But it is easy to show this directly. If we apply the second of 

 the equations representing Mr. Cayley's hypotheses to the par- 

 ticular case in which p^ = l, p^=Oj a case perfectly consistent 

 with the character of the original data, it will be found to lead 

 to the equation c^c^=0, an equation not implied by those data 

 in the particular case contemplated. On the other hand, I 

 afiirm without hesitation that there is no case in which the equa- 

 tions deduced by Mr. Wilbraham from my method of solution 

 can be proved to be erroneous. They do not, indeed, represent 

 "hjrpotheses,^* but they are legitimate deductions from the general 

 principles upon which that method is founded, and it is to those 

 principles directly that attention ought to be directed. 



I would request your readers to observe that I do not offer 

 the above remarks as affording any proof that the principles 

 upon which my method is established are true, but only as con- 

 clusive that Mr. Wilbraham^s objections against them, drawn 

 from what to him appears to be the anomalous character of an 

 equation to which they lead, are of no value whatever. Nor is 

 it difficult to see what is the source of the erroneous judgements, 

 for erroneous I cannot but term them, which Mr. Wilbraham 

 has been led to form. It is in a principle, the influence of which 

 appears to me to tinge the whole course of his speculations, that 

 those events which in the language of the data appear as simple 

 events, are the ultimate elements of consideration in the problem. 

 These are the elements in terms of which he expresses his equa- 

 tions, overlooking the fact that it is by mere convention that such 

 elements are presented as simple, and that the problem might 

 have been expressed quite otherwise. It cannot be too often 

 repeated that the distinction of simple and compound is wholly 



