On the Magnetic Declination at the Magnetic Equator. 3887 
ceding and succeeding observations. General Sabine 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 /, derived from the 
complete series of x observations; let m' be the mean derived from. 
n—1 observations, one observation o being accidentally lost ; then 
,__2m—o 
m= > 
n—1 
es m! —o , m—o 
m=m'— =m — . 
n 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 0+, we have 
“i nm+ x 
. nu 
lene 
> 
m—m 
. The comparative errors of m! and m" are therefore 
aS aT sa 
. 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° (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 o408 = 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, 
we shall find for the same month that the mean value of x=0-005 in. 
‘ 2 0°005 i vi 
nearly ; whence the error > =—5--=0:0002 in. only, and the error 
would never exceed 0°001 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 
m observations for each of two successive hours are each equal M, 
and that the observations for the same hours of the nth day are 
h : “ite 
respectively m'+/ and m'+/+.2, where m= uthe, Zis the limit 
beyond which observations are ‘rejected, and # is the excess of the’ 
observation to be omitted. The means retaining all the observations: 
2C2 
