348 G. H. KNIBBS. 
Not only will the difference of the refractions in the two 
observations involve the directions being determined for slightly 
different altitudes, the same consequence will also follow from the 
fact that the level and other corrections will not be the same; 
and further a satisfactory observation at the right moment will 
often fail to be made either from want of skill, from the presence 
of clouds, or from other causes. The difficulty in this respect may 
be obviated either by observing the altitude and azimuth both 
before and after the observation employed, and using the three 
results for the small interpolation from the middle one to the 
proper value of the altitude.t In this instance the observed 
values may be corrected before the application of the various cor- 
rections. Failing such additional observations the corrections 
must be made by spherical trigonometry. 
Let Z PS denote respectively the zenith, the elevated pole, and 
the star in a celestial triangle, the parallactic angle q subtended 
by the colatitude, being at 8S. Then reckoning the azimuthal 
angle A from the elevated pole as positive either way the small 
azimuthal correction dA, for a small difference of zenith distance 
dé will be expressed by the formula 
d 
dA = — df cosec fco Be Vive ve 48 
{ cosec { cot (¢ + — 5) +--+ ) 
dp denoting, with its proper sign, the variation of the polar dis- 
tance in the time dé, both being expressed in arc. Dividing by 
arc 1” reduces the circular measure dp/dt to ordinary angular 
measure. The value of the small term to be added to g, can how- 
ever never exceed 5} 5, or say 4’, and may ordinarily be neglected, 
since the correction dA is itself very small. 
Let ¥, p, A’ and 7’ denote respectively the colatitude Z P, polar 
distance ZS at apparent noon on the day of observation, one half 
the azimuthal angle 8,ZS, between the observed positions of the 
sun’s centre, so taken as to include the elevated pole, and one half 
of the elapsed time S.PS, between the observations ; then it will 
Dee eB a dae ee nc ey 
1 Three observations admit of a parabolic interpolation. If, howeveT 
_ the correction is very _— two observations will be ample. 
