NOTICES AND ABSTRACTS 
OF 
MISCELLANEOUS COMMUNICATIONS TO THE SECTIONS. 
MATHEMATICS AND PHYSICS. 
¢ 
᾿ MATHEMATICS. 
On a Question of Probabilities which occurs in the use of a fixed Collimator 
for the Verification of the Constancy of Position of an Azimuth Circle. 
By the AsTRONOMER Roya. 
Tue author said, his chief object in bringing this communication before the Sec- 
tion was to obtain the assistance of its mathematicians in extricating him from a 
difficulty, or at least a doubt, in which he found himself involved. In determining 
‘an azimuth, say of the moon, or of any other object, there were three independent 
elements which he used, and the probable errors of which he wished to compare with 
each other. There was first the assumed fixity of the circle itself, when its zero of 
azimuth was onve determined; there was next the indication of a fixed collimator, 
which was used as an independent check when its position in azimuth was once 
determined ; and thirdly, the daily observations of stars were used as a means of 
obtaining the required azimuth. The object was to determine and compare the pro- 
bable error of the zero of azimuth determined by each of these elements. Suppose 
now that the zeros of azimuth determined by these three elements on any one day 
were affected by the respective errors x, y,%. The results for determined azimuth 
of the moon on that day would be affected by the same errors 2, y, 2; but the com- 
parison of these results would not give us the numerical values of x, y, x, but of their 
differences x—y, r—z,y—x. These data however would suffice to give us the most 
probable values of x, y, x for that day, if we had their probable values X, Y, Z. 
xX Y Z 
dition, in addition to the equations given by the comparison of results above men~ 
tioned, furnished three equations to give the most probable values of 2, y, x for that 
day. But as the probable values X, Y, Z are yet unknown, the most probable values 
x,y, 5 for that day must be expressed by means of the symbols X, Y, Z. Now 
taking a series of such symbolical expressions for x, on a series of days, and treating 
them by the ordinary rules of probabilities as if they were observed errors, there was 
no difficulty in determining, from that series of errors, the probable error, still sym- 
bolically expressed ; and making this equal to X, an equation was obtained between 
X, Y, Z. In a similar way, by treatment of the values of y and of z, two other 
1850. B 
We should only have to make the sum (&) + (5) + Gy minimum ; and this con- 
