GALBRAITH’S ASTRONOMICAL OBSERVATIONS. 
655 
cos. 
since tan, cos. == R3 = 1 . But == cos. therefore 
sin. 
5 cos. w. V) X sin. 1" ^ w 
^ X = — ^2 sin. l"^tan. to X X 
12 sin. w cos. 2 zp sin. w cos. w 
and since the sin. 2 w = 2 sin. vj cos. we have 
2 sin. T' a? ^ w sin. 2" 
^ cc = — . . . . ( 12 ) 
sin. 2 sin. 2 w 
Taking ^ w — I’ , substituting for sin. 2 w its value when w =: 23® 27’ 40", 
formula (12) will become 
^x = 0-0000132748a? . . . (13) 
Log. of 0’0000132748 is 5-1230279 
By this means the correction for the variation of w from 23® 27' 40" may be readily 
obtained, by adding this constant logarithm and the log. of § tc in the given case to the 
sum of the logs, under I, the sura will be the log. of the correction of x. 
Example 1. Let w = 23® 27" 43' *76, ^ = 60“ § 3' *76, required the re- 
duction to the solstice. 
I. 11. III. 
Const, logs. . . 9-8555770, 2-745874, 8-03267 
A = 60m log. A ^ = 3-5563025, a ^ = 7-112605, a® = 0-66891 
1 — 4 . 43 1"-54 log. 3-4118795 9-858479 8-70178 
2- — 0-72 C. L 5-123 2d = — 0‘'-72 3d == --O' -05 
3= — 0-051og.^;^0-575 
4= + 0-13 9-110 
-I- 43 0-90 4th=*-{. 0"-13 
Cor. — 
00 OOOOiL(«h4h^ 
<J 
ooooioooomo 
When A does not exceed 30 or 40 minutes, which will in general be sufficiently dis- 
tant from the solstice, the operation by the formula, even in natural numbers, becomes 
remarkably simple, because in that case, the second and third terms are insensible. 
To render the first term applicable to every case, the sum of parts 11 and HI may 
be taken from the small table in the margin, and is always to be subtracted. 
Example 2. Let the sun’s right ascension be 7^ 16“ 36^ , the obliquity of the 
gcliptic 23*^ 27' 32"-8 and, consequently, A = 36s ^ ^ w = 7 '-2, required the 
reduction to the solstice ? 
In this way, the computation assumes the following very simple form : 
Const, logarithm. . .. .. .. ., .. 9-855577 
A = 111 16“ 36s = 76“-6, log. X 2 .... .. — 3-768458 
lstcor.== + 1« 10 '7''-6l0g.. .. .. .. .. .. 3-624035 
2d cor. = — 2*2 from this small table, ^ w ~ — 7' *2 log 0'857 
3d cor. = 0-4 ^ X from calculation. Const, log. 5-123 
x:=:-^ 1 10 5-0 = red. to solstice. ^a?log, . .. —9-604 
Hence, it appears that by this formula, the reduction to either solstice is a very easy 
operation. From these preliminary formulae it is now proposed to show their general 
application to one day’s observations, consisting of six sets or three pairs, made on the 
5th of July last, at Edinburgh, in latitude 558 57' 15 '•67.N, 
