316 G. H. KNIBBS, 
seconds of time multiplied by 15, before being squared. The 
latter equations are the more convenient, and the corrections are 
readily tabulated. In the following example, which will illustrate 
the precision of the correction method, the latitude ¢ is assumed 
to be 35°, the polar distance is 90°, and the zenith distances 44° 
and 46°. From these data we find, a =1°57'53-95", B=1°, t=V" 
42’ 41-70". By calculating in the ordinary manner and also from 
the “corrected” mean altitude! we get respectively, A denoting 
azimuth, and 7’ hour angle, 
Means A =134 30 39°71 T 3017 28-70 15416 9 
By (8)and (9) = 134 30 39-73 30 17 28-77 By (g) 54 16 25 
These differences are very small, notwithstanding that the case is 
an exceptional one, the interval between the observations being 
no less than 13™ 41-568, 
5. Error of the mean of true altitudes as a datum for the com- 
putation of azimuth or time: any declination. In general it is 
necessary to apply the corrections however, for cases were the star 
or sun moves in a small instead of a great circle, i.e. when its 
polar distance is other than 90°. As previously remarked, the 
line RQT in Fig. 1 will, in the general case, represent a great 
circle drawn through the positions occupied by the star at the 
moments of observation. The vertical ZH is drawn, not through 
the middle point of the star’s path as in the preceding section, 
but through the middle point Q of this great circle. It is evident 
from considerations of symmetry that the declination circle pass 
ing through this point will intersect the star’s path at its middle 
point S or 8’, hence to the formule already obtained it is neces 
sary only,to add terms depending on the distance of this middle 
point from Q. 
By Lagrange’s development, rejecting as negligible the powers 
of ¢ higher than the second, since they are of the same order as 
the similar terms rejected throughout, we have 
1 The corrections to the zenith distances are by (8) 2' 3°’48: by (9) 
1’ 0-64. 
