CHAP. XXI.] PROBABILITY OF JUDGMENTS. 391 



9. Before applying the general solution (15) (16), to the 

 question of the probability of judgments, it will be convenient to 

 make the following transformation. Let the data be 



#! = C l .... X n = C n , 



namely, 



Prob. Xi = i .... Prob. X m _ 2 - m - 2 ; 



and let it be required to determine Prob. X m . l9 the unknown 

 value of which we will represent by a m _ i . Then in ( 15) and (16) 

 we must change 



X into X m . } , Prob. X into m -u 



X m _i into X, n .<2 9 ttm-\ into a m -z* 



X m into X m .i + X m , a m into ,_! + ,; 



with these transformations, and observing that (X m .iX r ) = 0, 

 except when r = m - 1, and that it is then equal to X m _i, the 

 equations (15) (16) give 



' 



-A-i -^-m-2 -^-JM-I + 



Now from (19) we find 



X m _i ^ X m ^ X m .\ 4- X m 

 OT -i m a m -i + a m 



by virtue of which the last term of (20) may be reduced to the 

 form 



a m _l (XjXm.i) a m (Xj Xm 



X m 



With these reductions the system (17) and (18) may be replaced 

 by the following symmetrical one, viz. : 



i , a * * ^im 



X V Y ~ i% ^ ^ 



-^l ^2 -ILm 



These equations, in connexion with (7) and (13), enable us to 



