JUPITER WITHOUT HIS SATELLITES. 
253 
Sat. 
Sidereal Revolution 
Same in Seconds 
Sidereal Motion 
per Second 
Distance from 
Jupiter’s centre 
d. h. m. s. 
u 
miles 
1 
1 18 27 33-505 
152853-505 
8-478706 
278,542 
2 
3 13 13 42 040 
306822-040 
4-223947 
442,904 
3 
7 3 42 33-360 
618153-360 
2-096567 
706,714 
4 
16 16 32 11-271 
1441931-271 
0-898795 
1,242,619 
It will be observed, at once, that the period of the second 
satellite is almost exactly double the period of the first, and the 
period of the third almost exactly double that of the second; 
and, of course, a corresponding relation holds amongst the 
sidereal motions of these bodies. This of itself is remarkable, 
but far more singular is the relation which regulates the extent 
to which the above relations differ from exactness. It is to 
exhibit this that I have added the column of sidereal motions, 
because the relation in question is masked when the sidereal 
periods only are given. It will be found that the sidereal 
motion of the first satellite, together with twice the sidereal 
motion* of the third is exactly equal to three times the sidereal 
motion of the second satellite. Thus : — 
( 8 "- 478706 ) + 2 ( 2 "- 096567 ) = 12"-671840 = 3 ( 4 "- 223947 ). 
To show the effect of this singular relation, suppose the first 
and third satellites to start from conjunction, then after four re- 
volutions of the first satellite, the second has performed nearly 
one revolution, so that they are very nearly in conjunction again, 
but have in reality passed their conjunction by a small angle. 
At the actual moment of conjunction, the first has described 
three complete circumferences and an arc (A, suppose), w T hich is 
nearly a complete circumference, while the third has described 
the arc A only; thus twice the motion of the third satellite 
added to the motion of the first gives us three complete circum- 
ferences, and three times the arc A ; and therefore by the above 
relation the second satellite has moved through one complete 
circumference together with the arc A. Hence neglecting com- 
plete circumferences the actual change of position of each of the 
three satellites is the arc A, very nearly equal to a complete 
circumference. They therefore hold the same relative position 
at the end as at the beginning of the interval considered. Now 
nothing was said as to the position of the second satellite. As 
a matter of fact when the first and third satellites are in con- 
junction the second is always in opposition to both. Thus the 
actual changes of position are those exhibited in fig. 1, in which 
it is to be understood that the dimensions of the satellites are 
largely exaggerated. 
Wargentin, who devoted a life to the examination of the 
