MATHEMATICAL SECTION. 99 
If we express s and r in seconds of arc, and assume that the orbit 
is circular, = will be the semi-major axis of the apparent ellipse 
described by the satellite, and = cos h will be the semi-minor axis. 
age. Sant 
The quantities PRs cos h, H and & can be tabulated, and equa- 
tions (10) furnish the easy method of computing s and p which is 
employed by Marth (Monthly Notices, Royal Astronomical Society.) 
For computing & we have from the triangle TZO 
snAsink= cos(a—WN)sinJ 
sin h cos k = — sin (2— NV) sinJsind—cosJcosé (11) 
and, also, sinh sin k = — cosf 
sin h cos k = — cosg 
In what precedes it is assumed that the orbit of the satellite is 
known. If this orbit is not known the easiest method of proceed- 
ing seems to be the following: First, we assume the orbit of the 
satellite to be a circle, and from the observed angles of position 
and the observed distances determine the major and minor axes of 
the apparent ellipse described by the satellite around the planet, 
and the angle of position of the minor axis. Generally these 
quantities can be found by a graphical method. The preceding 
angle & is the angle of position of the minor axis, and cos h is found 
from the ratio of the two axes. Then from the triangle TOZ we 
have the equations 
sin J cos (N— a)=sin hsin k 
sin J sin (N — a) = cos h cos Ase sinhsindcosk (12) 
cos J = cos A sin 6 — sin h cos 6 cos k 
With the approximate values of J and NW found from these equa- 
tions we can compute the auxiliary quantities depending on the 
position of the plane of the orbit and the position of the planet, 
and can determine the elements belonging to the plane of the orbit. 
These approximate elements can afterwards be corrected by equa- 
tions of condition or by other methods. 
In work of this kind it is more convenient to have the inclination 
and node of the orbit referred to the equator, and since these ele- 
ments are commonly given with respect to the ecliptic we have to 
transfer them to the equator. If n and i are the node and inclina- 
