480 



we shall find for the same month that the mean value of #=0'005 in. 



nearly; whence the error -= ofi =0*0002 in. only, and the error 

 n .40 



would never exceed O'OOl in. A similar though less advantageous 

 result will be found in all classes of hourly observations. 



In the case where observations are rejected which differ from the 

 mean for the corresponding hour more than a given quantity, let 

 us suppose, to simplify the question, that the sums of n 1 out of 

 n observations for each of two successive hours are each equal M, 

 and that the observations for the same hours of the rath day are 



respectively m' + l and m' + l+tf; where *'= ^| 1 ' IS ^ e ^ m ^ 



beyond which observations are rejected, and x is the excess of the 

 observation to be omitted. The means retaining all the observations 

 are 



f i ' "W 



n " 



but if we reject the observation m 1 -\-l-\-x, we have 



m'= = m' 



It is assumed that z/ w a ' = (any other hypothesis of variation 

 would give the same final result), and therefore the error of the 

 change from the first hour to the second, when all the observations 



are retained, is - ; but if the observation be rejected, the change is 



m ( -\ m'=. 



n n 



This error, therefore, will be greater than the other if I >x ; so that 

 the error in the resulting change from one hour to the next will be 

 less by retaining an observation than by rejecting it, if the difference 

 from the preceding observation be not greater than the difference 

 from the hourly mean ; that this will most frequently be the case 

 will be obvious from the following fact : At Makerstoun, in 1 844, 

 at 1 A.M. the number of observations which exceeded the monthly 

 means by 3' and less than double that, or 6', was 99, while the whole 

 number which exceeded by more than 6' was only 16. 



It will be evident also that the difference / of an observation from 

 the corresponding hourly mean may not be due to irregular causes, 



