98 PHILOSOPHICAL SOCIETY OF WASHINGTON. 
These six auxiliary quantities are therefore strictly analogous to 
those which Gauss introduced for computing ‘the position of a 
planet. For controlling the computation, we have 
sin g sin h sin (H —G 
tang J = ; sin f oF , 
an equation in which each of the six auxiliaries enters into the 
value of J. 
If we introduce another auxiliary quantity, and put the ante 
TZO = 180° — k, 
it follows, from the manner adopted for counting an angle of 
position, that 
TZO = 180° — (p — k). 
Denoting the angle between the radius vector and the axis of 
Z by «, the spherical triangle TZS gives 
sin o sin (p — k) = cos(H + u) 
sin ¢ cos(p — k) = sin (H+ u) cos h (8) 
COs ¢ = sin(H+ u) sinh 
But we have also 
p’ sins = rsine 
p’ coss =r cosa + p, 
and by uniting these equations with (8), we can findsand p. This 
method of finding the distance and the angle of position is due to 
Marth, and as it is in constant use by him for the very convenient 
ephemerides of satellites which he publishes, it may be well to con- 
sider it further. If we multiply equations (8) by 7, and then sub- 
stitute the values of 7 sin ¢ and 7 cos ¢ from the last equations, 
we have 
p’ sin s sin (p — k) = 7 cos (H+ u) 
p’ sin s cos (p —k) = rsin (H+ wu) cosh (9) 
p’ COs 8 =rsin (H+ u)sinh+p 
Instead of these exact equations we may use in nearly all known 
cases of satellites the first two equations and put p for p’ and s si 
sins. The equations for use are then 
ssin (p — k) = 7, c08 (H+ u) (10) 
3 cos (p — k) =~ sin (H + w) cos h 
