312 G. H. KNIBBS. 
altitude increases. In order to get a definite idea of their magni- 
tude, we may suppose / to amount to 45° as a maximum in the 
class of observations to which we are referring, and dh to be 1’; 
then if c and 7 were even as large as one minute of are, the cor- 
rections would amount to only 1-5 and 2-1 respectively. It is 
evident therefore that the second differences of the corrections 
are extremely small, and consequently that the error of using the 
mean altitude, as the argument for the computation of a correction 
for the collimation or level constant, cannot lead to sensible error 
so far as least as the defect of instrumental adjustment is con- 
cerned. It may easily be shewn that the respective corrections 
€m and e€; on the correction applied to the mean altitude, the total 
difference being the altitudes being 28, are 
€Em= $e tan? B sec h (1+2 tan? h)+ ete......... (5) 
«, = ttan? B tanh (1+ tan? h)+ ete......... (6) 
which give for 8=1°, c and i=60", and h=45° the values, 0°039 
and 0-036 respectively, and even for h=60°, only 0-128 and 
0-127. With large theodolites and altazimuth instruments, the 
errors ought not to amount to even one fourth of this. Again, 
the interval of time between observations differing 2° in altitude 
can never be less than 8 minutes, and this is more than sufficient 
for the reading and recording of an observation and the prepara- 
tion for the next one. With half the interval the corrections are 
clearly but one-fourth of the amount. The above dictum is there- 
fore justified, and with this the discussion of the instrumental 
theory may be dismissed. 
3. Almucantars and great circles tangent thereto.—Turning to 
the astronomical conditions of the problem, it will be necessary 
in the course of their examination to determine the magnitude 
and law of increase of the distance between small and great circles, 
as for example between an almucantar or parallel of altitude, and 
a great circle of the celestial sphere tangent thereto, as we PTO 
ceed along the latter from the tangent point. By developing the 
tangent cone, whose line of contact with the sphere is the almu- | 
cantar of the zenith distance z, and employing the binomial 
