170 Dr Pearson, On a series of lunar distances. [Oct. 21, 



It is of course usual in working out a result from a Lunar, to find 

 by it the Longitude in time of the place of observation : in this 

 paper, on the contrary, the task has been, as the Longitude is 

 known, to deduce the error of observation: this is strictly speaking 

 the error in arc, the error in longitude depending on this, if it is a 

 real error in the instrument, and probably, in any case. 



It is impossible, in a brief article like the present, to give the 

 elements of each separate observation included in the series, though 

 it is hoped that they may be in the press before long. A general 

 analysis however of the results can easily be given : classing them 

 in groups of about 40 each, and considering the first of these only 

 at present, it was found that there were 29 cases in which the 

 measured distance was in defect of the theoretical distance, and 

 12 in which it was in excess. Assuming the rule given in p. 417 

 of the paper referred to to be correct, this result agrees exactly 

 with what might be expected : it being almost always most con- 

 venient, especially for a beginner, to take Lunars, especially from 

 the Sun, under such circunistances as will give this result : while 

 the example of India, founded on observations made at Madras, 

 seems to imply this probable facility, supposing that they were 

 mostly taken from the Sun on the new Moon, these being as easily 

 and certainly taken as any others in the northern hemisphere \ 

 In the four remaining groups, the proportions are 24 to 17, 

 28 to 15, 25 to 17, 17 to 14 : giving a total of 123 observations 

 in defect, and 75 in excess. Rejecting 3 or 4 certainly question- 

 able results, the greatest errors are 2' 59 in defect, and 2' 48" in 

 excess. In the first group the mean error in defect is 1' 18", the 

 mean error in excess is 0' 58". In the four remaining groups the 

 mean errors in defect are 1' 24", 1' 0", 0' 35", 0' 50"; while the 

 mean errors in excess are 0' 45", 0' 59", 0' 46", 0' 51"^ 



The mean error we obtain from the five groups taken together 

 is 1' 1" in defect, and 0' 52" in excess. The longitude of a place 

 established from these observations will therefore be in error to 

 the amount which will be indicated by the difference of these final 

 errors, or about 4^". This error in the measured distance implies 

 an error of about 9* or 2' 15" geographical longitude. As noticed 

 above, the error at Madras previous to the precision of recent 

 times, amounted to about three times as much, or 28^. The first 

 group however would have given an error twice as great, or about 

 4' 30" of geographical longitude in the same direction as that 



^ It is stated in Markham's Indian Surveys, that a series of about 800 (? sets 

 of) lunar distances taken from 1787 onwards had made tlie longitude of the 

 Observatory at Madras to be reputed in 1830 at 80" 21' 25" E. It is now fixed at 

 SO*' 14' 20", giving an error in former years of 7' 5". 



2 If two unusual errors of 3' 12" and 3' 47" be omitted, the means in excess 

 for the third and fifth groups will be 50" and 37" instead of 59" and 51". 



