88 A Method for ascertaining Latitude and Time 
Such, in fact, is M. Gauss’s tenth formula. 
3 BINT AN": 
It might be written, tang ¢= aia (sin ¢ tang h + cos 3 
cos uw). ‘ 
The trigonometrical method has given us above the simplest 
f : sin Asin (d-+2) 
ormula, sin ¢ = ——— : 
cos % 
Or, sin ¢ = sind sin kh + cos bcos h cos w. 
_The problem is then completely resolved. It may be 
simplified; but Jet us previously make a useful remark, in 
order to distinguish, beween the two solutions of the pro- 
blem, that which is applicable to the observations. 
The triangle to the pole between the two spots where the 
stars have been remarked, gives sin V: cos #:: sin 9: sin 
D = sin 3d side of the triangle. 
_ The triangle to the zenith gives cos h’: sin W:: sin D: 
sin{A — A’) = sin of azimuth difference. 
Sine ayy i sin D sin W __ ‘sing cos 0 sin W 
cof.» sin V cos hi * 
mre . (A—A) sin V cos i’ cos ht 
Or, sin W = sin 7s Rad can Ie and coy? 18 2 
quantity essentially positive. i 
Sin (A’—A) will always be positive, if you reckon the 
azimuths from one star to the other by the shortest distance. 
Theréfore sin W will have the same sign affixed to it as 
pane you will therefore know whether W is a positive or a 
negative angle. 
Which is the equation given by M. Gauss in order to, 
set all doubt aside :—there is no need of calculating it. It 
will also be easy without it to know the sign of (ey) 
’ sin § /? 
which is sufficient. 
If the two observations.are both to eastward, wu = W—V; 
if one is to eastward and the other to westward, u=V—W, 
unless (d AX — motion of rotation) be negative. 
In order to render the calculation of the angle V easier, 
; tang d 1 
M. Gauss makes (11) tang F = “arg *hee) coe 
cot F = cos 6 cot 6’; in which we perceive that his angle 
F is the complement of the subsidiary angle which I have 
named x. 
or, 
cosF tang§ tang §cosx 
G2) tae = sin(F—0) — cos (d+) ° ; : 
So for the angle V we may say that our formulas are identical. 
_ (13) cos 
