Satellites of the Georgium Sidus, See. 75 
being given, we may find an easy method of ascertaining the 
distance of the satellite, when it is near the apogee or perigee : 
for it will be sufficiently true for our purpose to use the fol- 
lowing analogy. Cosine of the distance of the satellite from 
the apogee or perigee is to the apogee distance from the planet, 
as the greatest elongation is to the distance of the satellite from 
the planet. When the ellipsis is very open, this theorem will 
only hold good in moderate distances from the apogee or peri- 
gee ; but, when it is a good deal flattened, it will not be con- 
siderably out in more distant situations : and it will also be 
sufficiently accurate to take the natural cosine from the tables 
to two places of decimals only. When this is applied to our 
present instance, we have ,91 for the natural cosine of 24,5 
degrees ; and the distance of the satellite from the planet will 
come out — * 33 » = 21", 8. 
,91 
By this method, it appears that the satellite, when it could 
not be seen, was nearly 22" from the planet. 
We must not however conclude, that this is the given dis- 
tance at which it will always vanish. For instance, the same 
satellite, though hardly to be seen, was however not quite in- 
visible March 2, 1791. Its distance from the planet, computed 
as before, was then only --~ - bL = 19", 8. 
The clearness of the atmosphere, and other favourable cir- 
cumstances, must certainly have great influence in observations 
of very faint objects ; therefore, a computation of all the ob- 
servations where the satellites were not seen, as well as a few 
others where they were seen, when pretty near the apogee or 
perigee, will be the surest way of settling the fact. The result 
of these computations is thus. 
L 2 
