OF TESTIMONIES OR JUDGMENTS. 619 
1—a,c,—a, e-2tD +) 
1a, pea 1) w+) a (¢+1) 
guy _ (@+1) (y+1) + aw(y+1) (s +1) 
¢ Lan tes r 
Whence we find 
a,(1—e,)a,(1—e,) (l—a, ¢, —a, ¢,) _ vy (20) 
(1—a, ¢,) (l—a, ¢,) Was 5 ; 
By means of (18), (19), and (20), we reduce (17) to the form 
2 i 1~¢,) a,(1—e,) (l—a, ¢, —a, 
w= Ba 1%P1 + ee diy 0 Py + 0! a ae = ue aay a) 
therefore effecting a slight reduction 
eae { - =e Py + ee CoP + c(1—a, ¢,—a¢2) } (21) 
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 s and ¢ 
their values, and reducing as before, we find that c is the probability that if xy w v 
take place, wyz will take place. In the end this amounts to the following state- 
ment. 
¢ = 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. «yz by changing 
Pp, p, and ¢ into 1—p,, 1—p,, and 1—e. If we effect this change, and then substi- 
tute the expressions above found, in the formula, 
Prob, zyz= 
Prob. xyz 
Prob. xyz+ Prob. xyz 
We shall find 
l—a,c l-—a,e 
Prob. xyz _ Tae peeps Tage Pa He (ba 4465) 
Prob. zy l—a, ¢, 1l+a,c, 
{= ¢, + =~—*e, + 1—a, ¢, —a, ¢ 
lc, 1 = 2 1% 2 C2 
(22 
1—a,¢, l-a,c, ) 
Toe as €y Py te(1—a,c,—a, ¢,) 
ene aS ee ee ee ee 
1—a, , , 1-4, 
d—e¢,.4) Ye, 7 
1+ 
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 
