by means of Two known Stars. 87 
sin V 
cos A sin 9 cos 0 
(sin A’—sin Asin 3 sin ¥ —sin h cos 6 cos 9 cos 0) ......8: 
and w= V—e(V—u),oru= (u—V) 4 V. 
It is not easy to perceive, at*first sight, how M. Gauss 
has been led to seek the auxiliary angle V, by means of the 
formula (7); but the processes of the calculations we have 
given, show how the author may have been led to it. 
It remains to know whether the angle V might not be 
one of the angles of the two triangles we have made use 
of. I see first that the equation (7), which gives the value 
of it, may be put under a simpler form ; 
cos 6 tang 0” 
Cot ¥V = — .—;—— =sin 0’ cot §. 
sin 6 
Shall then have cos W = cos (V'—w) = 
Giang = 3 ee 
ae® " =~ cos 3 tang &'—cos § sin 6”. 
Therefore I now recognise the formula of spherical tri- 
gonometry, which gives the angle to the first star in the 
triangle of which (90 — 2) (90—0’) are the sides, and 6 the 
angle comprised ; and which thus has one of its summits at 
the pole, and the othe? two at the points of the two stars. 
The angle V is thus that which 1s determined by the two 
first analogies in the vulgar method. 
_ The difference of the angles W and V, of the formulas 
7 and §, serves then to show the valueof W=+4+W4V, 
or the angle to the first star in the triangle which has given 
rise to the equation (1). 
In this triangle we know the sides (Q0—/) (90—8), and 
the angle comprised w: we shall get the angle to the pole 
of the first observation by the formula 
cos h sin u 
mn a os (9) which 
cos 6 sin A—siné cos h sin % (9) 
sin u 
eould.be set.down, tang. A = —,---—,-- "= ‘ 
; cos ¢ tang A—sin ¢ cos u 
is the horary angle of the first star at the time it has 
been observed; if it was then east, AR first star — A = 
right ascension to the middle of the sky, from which you 
may conclude the time of the first observation, and the 
correction of the clock. 
If the star was to westward, AX first star + A = right” 
ascension to the middle of the sky, of which you will make 
' the same use. 
Lastly, the combination of equations (4) and (5) will 
ive you tang @ = sina pint ee sin h + cos cos h cos vy, 
£ ) Pi cos A sin a i 
Puivgis (10). Such, 
Tangent A = 
