194 PROCEEDINGS OF THE AMERICAN ACADEMY 



Shortly after this investigation of Gauss, Bessel examined long series 

 of observations, to see whether the law of distribution of errors thus 

 indicated was a true one : he found that it was approximately so. His 

 tables and some results are in the Fundamenta Astronomic : from 

 these it appears that the definite integral above mentioned does rep- 

 resent the actual distribution of errors with striking exactness ; but 

 that there is generally a surplus of perhaps 1 to 3 per cent of the larger 

 errors. 



I may here mention that the least favorable series quoted by Bessel 

 — Bradley's declinations — are now in process of re-reduction by Prof. 

 Auwers, of Berlin, and that his results in part are in my hands for 

 anotlier purpose. The larger discrepancies which Bessel's own reduc- 

 tion left in them will pi'obably be found to disappear in the newer 

 calculations, and seem to arise from variations in the zero point of the 

 quadrant. 



A few years after Bessel's results were "published, Gauss wrote his 

 Theoria Combinationis Observationum. In this he takes the ground, 

 from the beginning, that 



e ^ 



does represent the probability of eri'or, and mentions casually that it 

 is only an approximation. 



About 1838, Bessel published his last paper upon this subject. 

 It seems to be little known in this country, but is extremely im- 

 portant. 



He shows that the law of error will be that mentioned with grreater 

 approximation, the nearer the following conditions are complied 

 with. 



First, that the sources of error are very numerous. 



Second, that they give rise to errors of equal average magnitude. 



He then points out that the first condition always holds good, by an 

 enumeration of the known sources of error ; and tliat in good observa- 

 tions the second condition has always a tendency to maintain itself, 

 because if any one source of error is sensibly more influential than the 

 rest, it will be detected and put away, or at least its effect diminished 

 by a proper arrangement of the work. 



The main object of this paper is to give the rules for good observing 

 derived from this theory : 1 have tested them in two long series of 

 observations, one made at Cambridge from 1862 to 186G, the other at 

 Chicago from 18G8 to 1871. The first series is of right-ascensions of 

 the principal stars, about 500 in number, each observed at least eight 



