CHAP. XXI.] PROBABILITY OF JUDGMENTS. 393 



though we ought not wholly to rely upon this hypothesis, we 

 cannot wholly dispense with it, or with some equivalent substi- 

 tute. Let us then examine the consequences of a limited inde- 

 pendence of the individual judgments ; the conditions of limitation 

 being furnished by the apparently superfluous data. From the 

 system (1) (3) let us, by the method of this work, determine 

 Prob. X m .i 9 and, comparing the result with (2), determine c. 

 Even here an arbitrary power of selection is claimed. But it is 

 manifest from Prop. i. that something of this kind is unavoidable, 

 if we would obtain a definite solution at all. As to the principle 

 of selection, I apprehend tha,t the equation (2) reserved for final 

 comparison should be that which, from the magnitude of its nu- 

 merical element OT _i, is esteemed the most important of the pri- 

 mary series furnished by experience. 



Now, from the mutually exclusive character of the events 

 denoted by the functions Xi 9 X 2 , . . X m .i 9 the concluding equa- 

 tions of the previous proposition become applicable. On account 

 of the symmetry of the same functions, and the reduction of the 

 system of values denoted by c* to a single value c 9 the equations 

 represented by (22) become identical, the values of x i9 x z , . .x n 

 become equal, and may be replaced by a single value x, and we 

 have simply, 



Jf = I? (4) 



a^xXi) a m (xX m ) 



The following is the nature of the solution thus indicated : 



The functions X 19 . . X m _i, and the values a^. . # OT _i, being 

 given in the data, we have first, 



From each of the functions X i9 X 2 , . . X m thus given or de- 

 termined, we must select those constituents which contain a par- 

 ticular symbol, as x i9 for a factor. This will determine the func- 

 tions (xXi\ (xX 2 ), &c., and then in all the functions we must 

 change x i9 x 2 , . . x n individually to x. Or we may regard any 



