OF TESTIMONIES OR JUDGMENTS. 645 



to the exceeding improbability a priori of their f ombination, the expression for 

 their united force tends to assume the form 



PlP-2 • • Pn 



PiP 2 • • !>»+(! -Pi) O--P2) ■ • (1-P») 

 commonly assumed to express the general solution. 



Stilly, When the probabilities are so far from being cumulative that we feel 

 that we ought to take a mean between them, the above formula is replaced by 

 the following, viz. : — 



[P1P2 ■ - *»»} » 



\P1P2 • ■ Pn }« + { O--P1) O--P2) ■ ■ (!-jO ] "~ 



Qthly, This formula takes, in reference to ordinary judgments, the place of the 

 arithmetical mean, with relation to the problem of astronomical observations, 

 both being expressions of a more general principle. 



40. It will probably appear to some of the readers of this paper, that I have 

 dwelt more upon questions of philosophy and of language, than it is usual to 

 do in mathematical treatises, and that I have also, in various parts, assumed 

 the office of a critic, rather than that of an expositor of original views. Respect- 

 ing the first of these points, I will only express a hope, that I have nowhere 

 in this paper entered into discussions that are not strictly relevant to the subject. 

 Upon the second, I have to observe, that the theory of probabilities is one in which 

 as it seems to me, the critical office is especially needed. I do not think that it 

 is likely to gain much advance from mere analysis. As respects the original 

 portions of this paper, it is my strongest wish, that they should be regarded chiefly 

 as materials for future judgment. Thus it is possible that the theory which I 

 have developed with reference to problems of which the elements are logical, may 

 be found to involve inconsistencies as a scientific theory, though I do not think 

 this likely to be the case. But whether that theory shall finally be accepted or 

 not, it is, I conceive, of some present importance, to establish the necessary de- 

 pendence of any theory, professing to deal with the same class of problems, upon 

 what I have termed the conditions of possible experience, — to show how those con- 

 ditions may be determined, and how they are to be applied. As respects the so- 

 called principle of the mean, applied in certain portions of this paper, it is open 

 to inquiry whether it in all cases leads to results possessing the characteristic 

 property, noted in Art. 38, and the decision of this question would materially 

 affect our estimate of its value. Lastly, it is, I think, highly probable, that con- 

 ditions which we do not yet know of may be discovered, affecting, not the possi- 

 bility of the data of a problem as discusse d in this paper, but their adequacy, 

 and the principles which, in statistical research especially, ought to guide us in 

 their selection. I am so conscious how limited, imperfect, and in some cases fluc- 



VOL. XXI. PART IV. 8 L 



