98 Prof. Challis on certain Questions 



does not "possess this characteristic. It may readily be shown 

 that that equation will always furnish a single root satisfying the 

 two conditions 



m;= 1— Ci(l— j9,) w^l — Cgfl—Pi) ; 



but that the remaining three conditions assigned in (10) will not 

 be satisfied by it unless certain other conditions, distinct from 

 the conditions of possible experience, obtain. 



And this leads me to notice, in the last place, a remarkable 

 distinction, a posteriori, between unwarranted hypotheses in the 

 solution of questions in the theory of probabilities, and axiomatic 

 principles flowing out of the very idea and definition of proba- 

 bility, or sanctioned by the laws of thought. It is that the latter 

 never impose, either upon the data or upon the solution, any 

 limitations but those under which alone experience is possible, 

 while the former do in general (always, I think, when the equa- 

 tion of a solution rises above the first degree) impose such limi- 

 tations. 



But these considerations only conduct us again to that general 

 view of the theory of probabilities which is contained in my reply 

 to the strictures of Mr. Wilbraham. They confirm, so far as 

 they go, the doctrine already advanced, that its claim to rank 

 among the pure sciences must rest upon the degree in which it 

 satisfies the following conditions : — 



1st. That the principles upon which its methods are founded 

 should be of an axiomatic nature. 



2nd. That they should lead to results capable of exact verifi- 

 cation, wherever verification is possible. 



3rd. That they should be capable of a systematic development 

 consistent in all its parts and processes, and neither acknow- 

 ledging nor imposing any limitations but those which exist in 

 the nature of things. 



Lincoln, July 6, 1864. 



XIV. On certain Questions relating to the Moon's Orbit, in reply 

 to the objections of Mr, Adams. By Professor Challis. 



To the Editors of the Philosophical Magazine and Journal, 



Gentlemen, 



BEFORE replying to the contents of the letter addressed to 

 you by Mr. Adams on my new theorems relating to the 

 moon's orbit, I think it right to state, in explanation of the 

 origin of the discussion, that Mr. Adams came to the decision 

 that the paper containing those theorems which I submitted to 

 the Cambridge Philosophical Society was without merit, and 



