134 



theoretically correct, or to be depended upon absolutely as a test of 

 the accuracy of the observation ; but it may perhaps be assumed 

 that any errors arising from this cause will not materially affect the 

 mean of a very great number of results. 



It appeared then that the mean of 218 differences taken at hazard 

 from among such as were most accessible, and from observations 

 made by different observers, was 3 7'' 6 7, and that the corresponding 

 mean of the means of all the probable errors was 13 ft 29 ; that is to 

 say, the latter is 35 -27 per cent, only of the value of the former. 

 As some proof that the cause, whatever it may be, is not very variable 

 in its operation, I may add that the first 110 differences, which were 

 all obtained before the end of 1854, give these numbers, 37'* 79, 

 ll'-86, and 31 '37 per cent, respectively. Again, the last 108 differ- 

 ences, which were all derived from the observations of one observer 

 only, give 37''56, 14 f '75, and 39 '27 per cent. Of these 108 last 

 differences, the first 50, taken from the middle epoch of all the obser- 

 vations, give 40'-06, 14 f< 60, and 36'44 per cent., and the last 58 of 

 the 108 give 35'-40, 14'-88, and 42-03 per cent. There is, however, 

 a circumstance which must be taken into account in making a com- 

 parison between the first 110 differences and mean probable errors, 

 and the last 108. During the course of the observations from which 

 the former were derived, it was the practice to take always 10 mea- 

 sures of each star on each night when possible ; during the observa- 

 tions from which the latter were derived, 6 measures only were taken. 

 This would tend to make the differences less in the former case ; and 

 with respect to its effect on the probable errors, if we put F for the 

 error of a single measure in each case, the probable errors in the 

 former case should be less by a quantity = 0'0768xF; for if we 

 put C for the constant and P for the probable error, then we have 



P= = xF, where 6 measures are obtained; and P= =xF, 

 \/45 -V/45 



where 10 are taken ; and the difference between these values 

 = 0768 F. 



The facts above disclosed create the difficulties in applying the 

 Calculus of Probabilities which have been before referred to. 



In the first place, the partial measures obtained on each separate 

 night are generally too few in number to eliminate the effects of one- 

 sided chance errors. 



