G. Hinrichs on Planetclogy. 279 
§ 15. The incommensurability of the periodic times. 
_ By the third law of Kepler we have, if T,; and Ty are the 
periodic times of two planets, 
Ty a} 
— = {| — i eo a eee 
m=) at 
ay 
ot by (88), Ty reeey!g 
| e_ Sobers 
ae Te Gey a ee ere ) 
_ which expression will not generally make Ty and T; commensur- 
_ ible, Thus we see that our law accounts for another important 
_ wndition of stability of the system, (see § 3, 1). 
but as the distances are continually decreasing, and at differ- 
tht rates, (this Journal, xxxvii, 41, gives the numerical values 
rates), we perceive that in time such commensurability 
_ ‘may take place between any two planets.’ Such is actually the 
_ ase between Jupiter and Saturn, as discovered by Laplace. 
The distances were (see § 18) for Jupiter a, = 520, for Saturn 
= 1000, giving for (68) the continued fraction 2(1,1,2.. ) 
having the approximations, 
13 
oo. Tres ° 
oar: T, approached originally to 5:2; now it is very nearly so. 
___for Venus and the Earth the original distances 70 and 100 
er the approximations, 
< 2 5 12 29 
a 2) et 
Whilst Airy has found the commensurability 13:8 or nearly our 
S17 [18 g=99 - 17°8]. 
= In the dunar systems such commensurability is common; and 
itis for the satellites of J upiter that Laplace demonstrated : the 
te Proposition, 7f such commensurability exists but approxima- 
. ie will become exact in time. : 
 ,.YINg seen that the change in distance produced by resist- 
Mee Will make the ratio approach commensurability, it therefore, , 
4 'S We stated before, will become rigorousl 
; : fom (68) we find easily that the ratio will be 2 if the dis- 
: the are in the ratio of 92: 1, or (by continued fractions) as 
__-**Pproximative fractions, 
: 12.3.8 19.39 
aa vy 3 iv iv 
< = tates that the 
bration ne ry x ft hysical Astronomy, London, 1852, p. 98,) states tha 
that it 
: jovial moons is “independent of the effects of a resisting medium,” 
for sf will be preserved pps a such medion. ye ie probelly 
. it would de 2 relative magnitude of the resistince an 
the urbation pend upon the re aga ‘ 
CéL, vol. vit, Ch.vi, § 15, We express the prop i general terms, 
aN. 
