178 On the Theory of Probabilities. 



Now there is nothing special in this reasoning. It is equally 

 applicable to every form of V, and establishes the identity of the 

 conditions of possible experience in the data of a question of 

 probabilities, and those of mathematical consistency in the final 

 processes by which the question is solved. 



Other remarkable consequences, to which I shall but slightly 

 advert, flow from these investigations. 



Thus it follows from them that the numerical value of any 

 probability determined by the general method will always, pro- 

 vided that the problem be a real one, satisfy those necessary and 

 , sufficient conditions of limitation, the mode of determining which 

 I have illustrated in the August Number of this Journal (1854). 



It follows hence, that two modes of procedure are open to us 

 for the limiting of the solutions of questions of probability. We 

 can either determine simultaneously the conditions of possible 

 experience and those of final limitation, by the special method for 

 this purpose developed in this Journal (August 1854), or, dispen- 

 sing with this preliminary inquiry, we can so order the process 

 of solution by the general method as to cause the different sets 

 of values of the quantities s, t, &c. to be distinctly evolved. 

 Then, if among those sets there be found one that is wholly 

 positive, the conditions of possible experience will be implicitly 

 satisfied, and the final value of the probability sought will itself 

 also satisfy the conditions of limitation to which the preliminary 

 inquiry would have conducted us. It is obvious that the former 

 is the easier mode of procedure. 



I would beg, in conclusion, to observe that I have in the 



analytical portion of this paper aimed at little more than to give 



an account of researches not yet quite finished. The general 



V s 

 proof of the properties of the functions K and —-, I hope at no 



distant time to be able to complete and lay before the public. In 

 the meanwhile I do not desire that they should be received with 

 any greater confidence than the verifications actually furnished 

 may seem to warrant. Of the function K it seems not impro- 

 bable that other and important applications may yet be made. 

 The conclusions to which these researches point is this, — that 

 the theory which they are designed to test is a rigorously con- 

 sistent one ; that the conditions of its mathematical validity are 

 identical with the conditions of possible experience. Now I 

 apprehend that this character is peculiar to the theory under 

 consideration. Upon no merely mathematical basis could a doc- 

 trine possessing such capacity of verification a posteriori rest, 

 because other elements than that of number are involved in the 

 inquiry. Some attention must be paid to the philosophy of Ian- 



