ALTAZIMUTH SOLAR OBSERVATIONS. 313 
expansion for expressing the distance between this curve and the 
plane development of the great circle, it is at once evident that 
the distance sought is 
dz = 4S? cot z — 1 S* cot® z + ete......... (7) 
in which S is measured on the great circle. If dz be required in 
seconds, S should be expressed in seconds, and the terms S? and 
S* etc. multiplied respectively by arc 1’, are* I” etc. The second 
term, even for a zenith distance of only 5°, amounts only to 0-018 
for the semidiameter of the sun. For 5° altitude the first term 
similarly amounts only to 0-19. 
4. Error of the mean of true altitudes as a datum for the com- 
putation of azimuth or time; zero declination. In § 2 it was 
remarked that the mean of the observed altitudes and of the 
directions of a celestial object, as given by the instrument in 
reversed positions was often employed asa basis for computations 
of azimuth, and it was also shewn in that section that the 
differentials of the corrections for the instrumental constants did 
not sensibly affect the result. The error of using the mean altitude 
requires consideration, however, also from the standpoint of 
spherical geometry. We shall suppose that the observed altitudes 
are independently corrected for refraction, and that the mean is 
that of the corrected altitudes. 
In Fig. 1 let ZH denote a vertical circle crossing a star’s path, 
which may be represented by RST, RQT, or RS'T according 
as the polar distance PS, PQ or PS’ is less than, equal to, or 
greater than 90°; so that for any one of these three cases, RQT 
Will denote a great circle on the celestial sphere. Then if these 
ares be bisected by the points SQS’, the great circle passing 
through them will also pass through the celestial pole P. From 
R and T draw the great circles Rr and Tt cutting Z H at right 
angles, and draw also the almucantars Rr’ and Tt’, so that r’ and 
t' are points on the vertical ZH of the same altitude as R and T. 
Let us further suppose the difference of the altitudes corrected for 
refraction of these points to be 28, and the angle of intersection 
between the great circle and the vertical, viz. RQZ to be te 
