314 G. H. KNIBBS. 
Fig. 1. 
Then, ¢ denoting the mean of the corrected zenith distances of R 
and T, we shall have by (7), 8 being supposed small, so that no 
sensible differences exist between the are, sine, or tangent of this 
angle 
tr = 2 B* tan* 7 cot (C- 6)... aes: (a) 
Sb = SP" tan 7 cot ((4+ 8)... i206 .(b) 
with abundant precision, the term in £* being negligible, for even 
if R and T differ in altitude 2°, this fourth power will be only 
about yisvsoo0- Taking the mean of these quantities therefore, 
we have for the distance A Q, A being the point whose zenith 
distance is ¢ i.e, the mean of the zenith distances of R and T, 
AQ = 3 P* tan’J cot ¢ + ete (c) 
The omitted term in B*, for B=1°, ¢ and J=45° will involve an 
error of only y+5: the higher powers are quite insensible. The 
simplest method of deducing an exact result is to correct the mean 
of the zenith distances ; which uncorrected would, of course, give 
the azimuth or hour angle of the point B, the altitude of which is 
equal to that of A. Hence in computations of time, the quantity 
(c) with the negative sign should be applied to the mean of the 
