OF TESTIMONIES OR JUDGMENTS. 



619 



_ Qc+l)(y + l) 

 i — a 1 c 1 — a 2 c 2 — ^ 



1 — a x c x 



(x+1) (y + l)+yv(x + l) (f + 1) 



1 <z 2 c 2 



_(x + l) (y + l)+^w(?/ + l) (g + 1) 



Whence we find 



(1-^cJ (l-« 2 c 2 ) X 



(20) 



By means of (18), (19), and (20), we reduce (17) to the form 



~^i C i^ + l-a lCl a * C * p * + G (l-a lCl )(l-a 2 c 2 ) 



w = 



1 (X 2 c 2 



therefore effecting a slight reduction 



Prob. xyz 



(l-c^c,) (l-a 2 c 2 ) 



f 1 — a, c, 



The arbitrary constant c, interpreted according to the rule, is the probability 

 that if the event xyw v s ^ take place, xyz will take place. Putting for 7 and 1 

 their values, and reducing as before, we find that c is the probability that if x y w~v 

 take place, xyz will take place. In the end this amounts to the following state- 

 ment. 



c = probability that if both observations are incorrect, a pointer directed at 

 random to the quadrant in which the star is situated will point below the star. 



The value of Prob. xyz will be obtained from that of Prob. xyz by changing 

 p x p 2 and c into 1— p v l—p 2 , and 1 — c. If we effect this change, and then substi- 

 tute the expressions above found, in the formula, 



Prob. xyz 



Prob. xyz + Prob. xyz 



We shall find 



1 — a x c, 



1 — a n c 



Prob. xyz _ 

 Prob. xy 



l_l ' c iPi + 1 _ 2 C 2 c 2-P 2 + cil-a^-a^) 



1— a, c, 

 1 — c x 1 1 ■ 



Cn "T" J- ~~ €t-i C-t — ~ Ctcy Cn 



1 + a 2 c 2 

 l-c 2 



!_g C lPl + l_ Co C 2P2 + C(l-g! C X - ^ 2 C 2 ) 



.. 1 — a. 1 — a„ 



1+ 1 l °l + 1 ~ C 2 



l-c x X l-o 2 ' 



(22) 



This expression involves an arbitrary constant c which we have no means of 

 determining. This circumstance indicates that those principles of probability 

 which relate to the combination of events do not alone suffice to enable us to com- 

 bine into a definite result the conflicting measures of an astronomical observation. 

 The arbitrary character of the final solution might have been inferred from 



