318 J. F. ENCKE ON THE METHOD OF LEAST SQUARES. 



a rare combination, but of high importance in the cultivation of 

 modern astronomy. 



Experience shows that in the simplest kind of observations, 

 and with the utmost care to avoid all circumstances which 

 may occasion error, continued repetitions of the same ob- 

 servations always give results differing somewhat from each 

 other. The causes of these differences are unknown to us ; or, 

 if we choose to ascribe them to the imperfection of our instru- 

 ments, and to the uncertainty of all the perceptions of sense, 

 at least their action cannot be subjected to calculation. We 

 may however assume, that in a certain kind of observation, 

 both the number of the sources of error, and the number of 

 combinations of which they are susceptible, remain the same ; 

 and also that the same combination, whenever it occurs, will pro- 

 duce the same error. If we knew the number of all the possible 

 combinations of the sources of error, and if we knew how often 

 those combinations which produce equal errors are contained in 

 this number, we should be enabled, by the calculus of proba- 

 bilities, to compute a priori how often a certain error ought to 

 appear in a given number of observations, and we could calcu- 

 late the probability that it would not appear more or less fre- 

 quently than a certain number of times. The causes being un- 

 known, we may, on the other hand, apply the calculation of pro- 

 babilities to the results of experiments ; or, from the number of 

 times that an error has actually appeared in a number of ob- 

 servations, we may infer how often it should have appeared ac- 

 cording to rule, and how often it would appear in future repeti- 

 tions. This application only supposes that the continued re- 

 petition does not bring in any new source of error. The num- 

 ber of the sources of error, and of their combinations, remains 

 wholly undetermined. 



By the pi'obability of a certain combination, or of all the 

 combinations which produce an error of a certain amount, 

 we understand the proportion which the number of such com- 

 binations bears to the number expressing all possible com- 

 binations. On this proportion the probability of an error A 

 will depend. If this probability (which is necessarily a func- 

 tion of A, and of one or more constants having reference to the 



