ALTAZIMUTH SOLAR OBSERVATIONS. 319 
now propose to determine the further error involved, where instead 
of so doing, the mean of the observed altitudes, is corrected for 
refraction. Since the refraction increases more rapidly than the 
zenith distance it is evident that the refraction for the mean of the 
observed zenith distances is less than the mean of the refractions of 
those distances. The difference is very small near the zenith, but 
becomes very appreciable near the horizon. At 45° it is about 
0-"01 for a difference of 1° in zenith distance between the mean 
and either observation and, as will presently be shewn, varies as 
the square of that difference. Let z + b denote the zenith dis- 
tances given by observation, so that 2 is the means of, and 6 the 
half difference between the two: let also r be the refraction 
corresponding to z, and 7, and 7, the refractions for the observed 
zenith distances. Then e being the correction to the refraction 1, 
we have for the mean of the true zenith distances 
BEE (11 Hg) HSH A Eee s eons (2) 
For small changes of zenith distance the refraction may be put in 
the form 
r=k tan z (m) : 
in which, since & varies very slowly with 2 excepting near the 
horizon it may be treated as constant. Hence substituting in 
this last’ expression, z + 5 for z, we obtain for the value of the 
correction 
ée=r tan? b sec? z+etc (15) approx. 
the sign of which is always positive; in other words the mean of 
the true zenith distances, is greater than the zenith distance com- 
puted by applying the refraction correction to the mean of the 
observed zenith distances. The above formula, however, is only 
Suitable above 20° altitude at which value its error is about 0-02. 
Between z=70° to z= 90°, it is unsuitable, because the variation 
in k, not taken into account, is not negligible, and appreciable 
terms are also neglected, When very great accuracy is required 
nothing is gained for low altitudes by the application of the cor- 
rection to the refraction for the mean altitude: it is more con- 
venient to correct each independently. On the other hand the 
following table will permit of accurate interpolations of the error 
