OF TESTIMONIES OR JUDGMENTS. 625 



the insufficiency of the remaining data, necessary in order to give to the a poste- 

 riori probability of z a definite value, the solution obtained when that a priori 



value is neglected should involve the symbol ^. The presence of this symbol in 



a solution always indicates insufficiency in the data. And herein, as it seems to 

 me, consists the reason why the mind, impatient of incertitude even while dealing 

 with the very science of uncertain knowledge, is led to seek escape from its doubts, 

 by calling in the aid, in some form or another, of that adventitious principle which 

 I have denominated the principle of the mean. I say in some form or another ; 

 for I can conceive of another form of the same principle connected more directly 

 with the idea of a limit than with that of a mean. Thus as testimonies which 

 are insufficient of themselves to produce a definite expectation may definitely 

 modify a definite expectation previously formed, we have suggested to us the idea 

 of that limiting state to which perpetual and independent repetition of the same 

 series of testimonies would cause the mind, whatever its starting point of expec- 

 tation might be, to tend. And as this limiting state would be one which a further 

 repetition would not alter, we should thus arrive in effect at the same solution 

 as is indicated by the principle of the mean, in its direct expression. 



28. I have extended the preceding analysis to the case in which three obser- 

 vations are to be combined, a case which, in connection with the previous one, is 

 sufficient to determine the general law. The result is what the preceding analysis 

 suggests, and may be expressed in the following theorem : — 



If n conflicting observations assign to the altitude of a star the respective 

 values p x p 2 . . p n ; if, moreover, a x a 2 . . a n are the antecedent probabilities that 

 the observations will be such as they prove to be with respect to those circum- 

 stances which determine their relative accuracy, and ^ c 2 . . c n their respective 

 probabilities of correctness to a mind acquainted with these circumstances, i.e., to 

 the mind of the observer after the observations have been made, then the most 

 probable altitude of the star will be 



1 — a. 1 — or, 



1 — a n 



' + l- Cn CnPn 



1 — «! 1 — « 2 



1-c/ 1 + l-c/ 2 ' 



+ l ~ an c 

 + l-c n Cn 



This expression admits of the same deductions as the one before obtained for 

 the case in which the observations are two in number, and in particular it leads, 

 when the circumstances of the observations are judged to be in all respects the 

 same, to the principle of the arithmetical mean expressed by the formula 



Pi +-P2 • • +Pn 



n 



29. I have remarked that the principle of the arithmetical mean has some 



VOL. XXI. PART IV. 8 F 



