ALTAZIMUTH SOLAR OBSERVATIONS. 317 
QS or QS’=tan? }¢ sin 2p =} 2? sin 2p, very nearly......(7) 
the simpler form being also given at once, mutatis mutandis, by 
(7). This quantity is negative when the polar distance is greater 
than 90°, therefore the direction Q towards S must be regarded 
as positive. To the previous corrections to the zenith distances, 
therefore, we must add the difference of zenith distance between 
Q and 8, which is! 
Qs or Qs'=} #? sin 2psin J=}4 at sin (sin Qp......... (7) 
the last expression being deduced by the ratio Q T/Tt=S Q/Qs, 
and containing only factors given by observation. By means of 
(4) the final formulz for the general case may be expressed either 
in terms of fB and a, or B and¢. In this way, we may reéxpress 
(j) thus :— 
Qs or Qs’'=4a sin (cos p v(a? sin? (+ 8?) 
=4¢ cos p v(t? sin? p— B*).........s0c000s (k) 
and from these last expressions, and those previously deduced, 
obtain, by simple addition, the general values of the corrections 
sought, viz. « for azimuth, 7 for time. 
€=1 — f? cot ¢ (10) 
n=4 a. Bih ¢ [cos p v(a? sin? ¢+ 8?) —acos Cleat kad 
Since ¢ sin p is always greater than £, see Fig. 1, the similar 
expressions, in terms of ¢ instead of a, may be written, using the 
auxiliary for brevity, 
w=} ¢? sin? p [cot p v(1 -2 cosec? p) — cot ¢]...(12) 
«=o—} B? cot ¢ (13) 
Y= O+4 B* COb Cis ccccensisvacteiecesunee eww (14) 
The term within the rectangular brackets in (12) is a factor in 
which the unit of B and ¢ is indifferent: these quantities may 
therefore be expressed in either degrees, minutes or seconds ; the 
other factor, and the @ terms in (13) and (14) are easily tabulated. 
As we are dealing with small quantities the computations may be 
readily made, and involve less expenditure of time than is involved 
in the calculation of two spherical triangles. Tables I. and II. 
1 Note that sands’ are not shewn in Fig.1. They would be the points 
determined by letting fall the perpendiculars Ss and S's’ on to Z H. 
