ALTAZIMUTH SOLAR OBSERVATIONS. 315 
zenith distances; for it is evident from the figure that at the 
mean of the times the star would have been at Q. This point, 
however, does not represent the mean of the directions of R and 
T, since the former is 28 nearer the zenith. Since Rr=Tt= 
f tan J, the difference of azimuth between D—the mean of the 
azimuths of R and T—and Q, viz. the angle Q Z D, not drawn in 
the figure, is expressed by the formula 
Angle QZ D=48 tan J [cosec (¢-- 8)—cosec ((+ )]......(d) 
By expanding the terms in the brackets, multiplying by sin ¢ and 
rejecting the higher powers of ( as inappreciable, the distance C D 
from the vertical through Q, of a point D, so taken as to be the 
mean of the azimuths of R and T is obtained 
CD=f* tan 1.cnt (cic... (e) 
Multiplying by cot J, we get the difference of altitude between 
this point and Q, viz. 
CQ = B* cot 6i..c%5 epee (f) 
Remembering that if a denote the change of azimuth for the 
change of altitude B 
tan J=~ sin ¢ very nearly...... (9) 
since « and £8 are supposed small, i.e. always less than say about 
1’, the corrections to the mean zenith distance, for a star or the 
sun when the declination is zero, are, for the computation of 
azimuth and of time, respectively 
© = -(AQ+QC)= — (fa? sin 2¢+ f? cot {)......-+ (8) 
Ne = -AQ = — da* sin 2¢ (9) 
By means of the approximate equation 
B* +a* sin? (= ¢ sin® p...... (i) 
P denoting polar distance—in this case 90°—the sine therefore 
being unity, (8) and (9) may be reéxpressed by substituting the 
Semi-interval of time for the semi-interval of azimuth : thus 
€ = —} cot ¢(¢? sin? p+?) (8a) 
= —4 cot ¢(¢? sin? p— B?) (9a) 
In these eis equations, if the corrections are required in seconds, 
a, B, and ¢ should be expressed in seconds, and the result multi- 
Plied by arc 1”, The time ¢ must also be expressed in are, or the 
