280 G. Hinrichs on Planetology. 
For Jupiter’s satellites we have a, =16, a, =10, or a, : a,=8:5; 
and a, : a,=10:7=8: 5°6, therefore we find the periodic time 
of the second moon twice that of the first, T= 27, an 
periodic time of the third twice that of the second, T',=2T,; 
hence Laplace’s famous relation between the mean motions, 
n,+2n,=38n,. 
In the system of Saturn similar relations obtain. At first we 
had (see § 14, B) for the distance of Tethys (IIL) and Mimas (1) 
the ratio 04 : 44=8: 6:6, while the duplication of the periodic 
times requires the ratio 8 : 5. But Mimas has approached Satur 
the most, and thus this proportion (now 5:4: 8°4=8 : 5-04) bas 
been brought about. 
For the fourth and second we had originally, Dione : Eneela- 
dus=6'8 : 4°7=8 : 5°5, or likewise sufficiently near 8 : 5 that the 
duplication of the periodic time should become almost rigorous.” 
The lanar world of Uranus is particularly noted for such 
duplications, from the fact that Schweigger, as early as 1814, on 
such grounds predicted the existence and gave the orbits of the 
two innermost moons of Uranus, which were discovered by 
sell in 1851. The coincidence is very remarkable, as will be 
scen from the following:* 
Schweigger,1814,  Lassell, 1851. 
Uranus, I moon, - - - 21767 days, 2°5117 days. 
Il - - 4°3534 4:1445 “ 
and the IV (or II of Herschel) having a period of 8°7068 days 
approximates to the further duplication of the periodic times. 
Also the period of III is about half the IV period, the forme 
being 5°8926 days, the latter 10:9611. 
Taking only the first two decimals we find by means of con 
tinued fractions the following approximations: 
441] 13.8 6°25 os 
Itolor —>-~>+ 3 3 = etc. 
Iv « Cl SL ae 
i 441 1? «9-10 -29" 29 
Vv “yr Eo = TT 7” 50’ 107 ete. 
* Herschel, Outlines of Astronomy, § 550. Lassell 
. * Schweigger; Ueber die ree h. der ersten Uranustrabanten durch 4 
Astronomische Nachrichten, 1852, No. 832. 
