94 PHILOSOPHICAL SOCIETY OF WASHINGTON. 
In the method which Bessel has employed for computing the 
position of a satellite, he has derived his formule by Lagrange’s 
method. Thus if @ and 0 be the apparent right ascension and 
declination of the planet at any instant, a’, 0’ the same quantities 
for the satellite, and if p and p’ be their distances from the earth, 
and if r be the radius vector of the satellite, and a and d its right 
ascension and declination seen from the planet, we have, by the 
method of rectangular co-ordinates, 
p’ cos 0’ cos a’ = p cos 6 cos a+ 7 cos d cos a 
p’ cos O sin o = p cos d0 sin a+rcosd sina (1) 
‘sin & =p sin 0 +rsin d 
p p 
If p and s are the angle of position and distance of the satellite 
with respect to the center of the planet, the spherical triangle 
formed by the pole of the equator, the planet, and the satellite 
gives us the following equations: 
cos s = sin 6 sin ” + cos 0 cos & cos (a — a) 
sin s cos p = cos 0 sin 0’ — sin 6 cos & cos (a — a) (2) 
sin s sin p= cos © sin (a — a) 
If N and J be the longitude of the node of the orbit of the sat- 
ellite on the equator, and its inclination to the equator, and u the 
distance of the satellite from the node counted on its orbit, we have 
cos d sin (a — N) = sin u cos J 
cos d cos (a — NV) = cos u (3) 
sin d ' = sin usin J 
These three sets of equations are fundamental, and are sufficient 
for the complete solution of the problem—Given the orbit of a sat- 
ellite to determine its apparent angle of position and distance. We 
have only to transform these equations, and, in order to ease the 
computation, to introduce, as Bessel has done, certain auxiliary 
quantities which depend on the position of the planet in the heavens, 
and the position of the orbit of the satellite with respect to the 
equator. These auxiliary quantities will of course vary with the 
position of the planet, and also from the slow changes that the node 
and inclination of the orbit undergo, but they can be tabulated 
easily. So far, therefore, as the practical solution of. this question 
is concerned there is not much more to be desired, but it is interest- 
