358 G. H. KNIBBS. 
for observations (ii.) and (v.) should of course be taken ; that is 
to say, the altitude deduced from (iii.) should be allowed to influ- 
ence the result as well as that from (i.), and the same in regard 
to (vi.) and (iv.) 
It is perhaps hardly necessary to remark—see (v.) § 18—that, 
where there is any uncertainty in the latitude, observations should 
be taken if possible at equal intervals of time before and after 
apparent noon, especially if great precision be desired. 
21. Conditions of precision and general remarks.—Owing to 
the great range of the sun’s motion in declination, viz. about 47°, 
and the consequent change in the ratio between the azimuthal and 
vertical components for any given zenith distance, at different 
times in the year, the value of solar observations, for the deter- 
mination of meridian, greatly varies. If the ratio of the com- 
ponents, i.e, dA/dé be denoted by &, and the uncertainty of a 
measurement of the sun’s altitude through instrumental and 
observational defects, together with the uncertainty of refraction, 
by dé; then the whole uncertainty dA of the azimuth from these 
causes alone, will be 
dA=k dé (60) 
Within the tropics & is never very large for great zenith distances, 
but for places outside it may in midwinter become considerable, 
so that it is not unimportant to estimate its value, when forming 
an opinion as to the reliableness of observations at any given time 
of the year. For example, when the sun has its maximum polar 
distance of about 113°, it crosses a vertical circle at an angle of 
35°, not before its zenith distance is 803° for latitude 35°, and not 
before that distance is 891° for latitude 40°. The corresponding 
values of & are the cosecants of these angles, viz., 1-01 and unity, 
so that the uncertainty in azimuth is identical with that 10 
altitude. Consequently even in latitude 35°, midwinter solar 
observations for meridian have not a high value as regards their 
accuracy. 
The limiting values of latitude, at which observations made at 
a given zenith distance, will give a particular value of & for any 
