CHAP. XXI.] PROBABILITY OF JUDGMENTS. 395 



ought, on solution, to give the same value of x as the equation 

 (4) or (6). Now this will be the case. For since, by hypothesis, 



we have, by a known theorem, 



Xi^X, _X m _X l+ Xi..+ X m 

 i 2 ' a m a, + 2 . . + a m 



Hence (7) becomes on substituting j for X &c. 



a mere identity. 



Whenever, therefore, the events x l9 x z , . . x n are really inde- 

 pendent, the system (4) (5) is a correct one, and is independent 

 of the arbitrariness of the first step of the process by which it 

 was obtained. When the said events are not independent, the 

 final system of equations will possess, leaving in abeyance the 

 principle of selection above stated, an arbitrary element. But 

 from the persistent form of the equation (5) it may be inferred 

 that the solution is arbitrary in a less degree than the solutions 

 to which the hypothesis of the absolute independence of the in- 

 dividual judgments would conduct us. The discussion of the 

 limits of the value of c, as dependent upon the limits of the value 

 of ar, would determine such points. 



These considerations suggest to us the question whether the 

 equation (7), which is symmetrical with reference to the func- 

 tions Xi 9 X Z9 . . X m , free from any arbitrary elements, and rigo- 

 rously exact when the events z l9 x 29 . . x n are really independent, 

 might not be accepted as a mean general solution of the problem. 

 The proper mode of determining this point would, I conceive, be 

 to ascertain whether the value of x which it would afford would, 

 in general, fall within the limits of the value of c, as determined 

 by the systems of equations of which the system (4), (5), presents 

 the type. It seems probable that under ordinary circumstances 

 this would be the case. Independently of such considerations, 

 however, we may regard (7) as itself the expression of a certain 

 principle of solution, viz., that regarding X\ 9 X 2 , . . X m as ex- 

 clusive causes of the event whose probability is x, we accept the 



