479 



essential in the first instance to obtain the result including all the 

 supposed irregular actions, and afterwards to eliminate these in the 

 best manner possible. 



In the discussion of the Makerstoun Observations I had substi- 

 tuted for certain observations, which gave differences from the mean 

 beyond a fixed limit, values derived by interpolation from pre- 

 ceding and succeeding observations. General Sabirie in his discussions 

 has rejected wholly the observations which exceeded the limit chosen 

 by him. The omission of observations accidentally or intentionally, 

 and the taking of means without any attempt to supply the omitted 

 observations by approximate values, require consideration. 



Let m be the true hourly mean for an hour h, derived from the 

 complete series of n observations ; let m' be the mean derived from 

 n 1 observations, one observation o being accidentally lost ; then 

 nm o 



M = 



r> 



n 1 



m o 



=m 



n 1 



If, however, we supply the omitted observation by an interpolation 

 between the preceding and succeeding observations, and if the inter- 

 polated value be o + oc, we have 



nm+oc 

 m' f = , 

 n 



m=-m" -- . 

 n 



The comparative errors of m 1 and m" are therefore 



o m _ x 

 - r and . 

 n 1 n 



We may for any given class of observation determine the mean values 

 of these errors. 



Example : At Hobarton, in July 1846, the mean barometer 

 for 3 h (Hobarton mean time) was 29'848 in., and the mean differ- 

 ence of an observation at that hour from the mean for the hour was 

 0-403 in. ; if an observation had been omitted with such a difference, 

 or for which o m=0'403 in., we should have an error in the resulting 

 mean of -~-=0'016 in., and the error might have been twice as 

 great had the observation with the greatest difference been rejected. 

 If we now seek the error of m", where the observation is interpolated, 



VOL. X. 2 L 



