by means of Two known Stars. 85 
value of this angle, we may make use of it to obtain other 
useful equations. We have, to be sure, the analytical ex- 
pression of the sine w and of the cosine w, but they con- 
tain the two unknown quantities ¢ and a; they cannot, 
therefore, serve but as preparatives. 
Previous to using them, let us draw from formula A 
general consequences which may he of service. Let A 
_ be thé pole, B the zenith, and C the angle to the star; we 
2 in ABsinA - i A 
shall have sin C= Be a (—**) Chin tap hens 
sin BC a cos h 
sin AC cos AB — cos AC sin AB cos A 
Cos C = ( ts ECD icity 
This last formula is general; it belongs to all spherical 
triangles, and expresses the relation between the three sides of 
any triangle whatever,.and two of its angles. Generally, sine 
ist side cosine 2d side—cosine Ist side sine 2d side cosine 
angle comprised = sine third side cosine angle opposed to 
Si aide Me! (D) ) Ane sine 2d side xd ing vie comprised angle 
sine third side 
= sin angle opposite to second side.....(E). - 
This formula, which we have just found by analysis, is 
the fourth of eighteen of the same kind I have found by 
synthesis, which I have explained in my Courses at the 
College de France: they are not of service in the common 
calculations of spherical triangles, because they require the 
knowledge ‘of four of the parts of the triangles, instead of 
three, which are sufficient in all cases; but they are useful 
in analytical disquisitions, and we shall have occasion to 
use them again in the course of this memoir. 
Supposing the angle to the first star or angle w is known, 
we shall conclude that sin g = sin & sin § + cos hcos 3 
cos u......(5); A and dé are the complements of the sides 
comprising the angle ~, 9 is the complement of the side 
opposite to angle w: here is then a mean to ascertain g, as 
soon as we shall have determined the value of the third un- 
known quantity that has been introduced, that is to say, of 
angle wu. 
M. Gauss finds analytically the equation (5) by the 
combination of his formulas (1) and (3); but we endeavout 
to reduce all analytical operations to known rules of spheri- 
cal geometry, in order to render the relation.or rather the 
identity of the two methods more evident. ~ 
. : : cos ¢ sin A : 
The equation (3) sn w= aa —, by simply re- 
: vor ; : cos h sin w 
versing the quantities, gives sin A = ——~ = 
i sin @ 
F 3 sin 
