136 



the squares of the errors found by such comparison, and m for the 

 whole number of nights, we have 



(m-1) e 2 ='4549xS, 



which gives the value of e. 



If A be taken to represent any convenient constant, the combining 



weight for each of the first night's observations will be 5 , for 



iii" ~\j 



A 



each of the second night's - - , and so on. 



Let P stand for the probable error of the final result, then 



1 



P= 



\/u - 



V+/ 2 n^ 

 The probable error of the mean of the first night's observations 



^-j, of the second= A / [ e*-\-l ), &c. 



Mr. Airy, however, while remarking that the mode of proceeding 

 above described is the only one which really meets the difficulties of 

 the case, admits at the same time that it would not be expedient to 

 use so elaborate a process in dealing with observations like those in 

 question, in which the ordinary errors of observation are large in 

 amount, and in which such extreme accuracy in the results is not 

 obtainable as in some other cases to which the principles of the 

 Calculus are applicable. 



He suggests therefore that all the observations of all the several 

 nights should be combined together for the purpose of obtaining the 

 probable error and weight of the final result ; and this may be done 

 in two different ways :-^First, by treating all the single measures of 

 all the nights, as if they had been made on one and the same night, 

 and obtaining the final result and its probable error and weight ac- 

 cordingly in the usual manner : Secondly, by treating each group or 

 set of 6 or 10 as a single observation. 



The only other method of proceeding is that above described as the 

 correct one, but which has not been adopted, as being too cumbrous 

 for the occasion. This will be designated as the Third Method. 



For the purpose of ascertaining the result of employing each of 



