654 
GALBRAITH’S ASTRONOMICAL OBSERVATIONS, 
By spherics, sin. k tan. w — tan. ^ = tan. (w x)* 
But tan. {w - — x) = 
tan, w — tan. x 
therefore, 
tan. w tan. x 
tan. w — tan. x 
sin. k tan. w ~ 
1 4* tan. w tan. x 
which by reduction becomes, 
(1 — sin. &) tan. w 
tan. X — .. .. ., (8) 
1 4* sin. h tan.2 vj 
This equation would give the reduction to the solstice, but it is not in a form to'be 
readily applied. It admits of a transformation, however, from the following considera- 
tions, which renders it remarkably simple. Since k does not in this case differ much 
from or 18h let — kr k — 18i> — k — 18^ , and ^ being small 
cos. fc = 1 4- . 
2 24 
stituted in formula (81 it becomes, 
, A® A* 
(1 _ 1 + 
2 24 
tan. X— — 
A** 
720 
A® 
4 — 
720 
&c. If this value of cos. k be sub- 
&c.) tan. w 
2 
H- 
(9) 
A'^ 
720 
&c.) tan. 3 w 
0*1883608, 
1 (_ 4 .. 
2 24 
Now taking w = 23^1 27' 40'', tan. iv = 0*4340056, and tan. 2 to 
By introducing these values into equation (9) it becomes, 
0*2170028 — 0*0180836 A^ 4* 0*0006028 a®^ 
tan, X — 
1*1883608—0*0941804 A^ 4“ 0*0078483 a'* —0*0002616 a® 
tan. 0 ;*= 0*18260684 a^ — 0*0007454 a^ — 0*00075777 A® (10) in which a is the 
length of the circular arc to radius unity. 
It is new only necessary to adopt the co-efficients of formula (10) to degrees of arc 
or minutes of time, as these are the terms in which the right ascension of the sun is 
generally given, while tan. x may in like manner be converted into seconds of arc. 
This is accomplished by applying the logarithms of R‘^, R", &c. to the logarithms of 
the co-efficients of formula (10), and they become those for a expressed in degrees and 
X in seconds. 
I. II. HI. 
Const, logs. 1*0596970, 5*154114, 1*64523, . (A) 
Similarly are obtained the logs, of the constants for minutes of time when the right 
ascension is given in time, and the distance from the solstice is known in minutes of 
time and decimals. 
I, II. III. 
Const, logs. 9*8555770, 2*745874, 8*03287* . ^) 
To render these co-efficients generally applicable, it is necessary to find the variatiom 
of X corresponding to a change of one second in w. 
For this purpose from formula (9) we get 
A^ tan. ic 
tan, X 
1 4*’ tan. 3 w 
5 
X — — A ® sin. 1" tan. w 
12 
Differentiating equation (H) and 
x‘~ 5 ^ w 
“ — A® ^ 
12 cos. 3 to 
— A ^ ^ nearly, and thence, 
12 
(H) 
— — A sin. 1" tan. co. cos. 
12 3 
® O”.7170955 a 2 - 0»Vo0000005570i4 A4 - &C. 
