THE ELEMENTS OF THE ORBIT OF A SATELLITE. 
877 
The complete investigation of this subject involves considerations which will require 
special treatment. In § 22 it is only so far considered as to show that, when there 
is identity of the periods of revolution of the moon and earth, the angular velocity 
of the system must be much greater than that given by the solution in § 18 of 
“ Precession.” 
When the earth rotates in 5-g hours, the motion of the moon relatively to the earth’s 
surface would already be pretty slow. If the system were traced into the more remote 
past, the earth’s rotation would be found getting more and more rapid, and the moon’s 
orbital angular velocity also continually increasing, but ever approximating to identity 
with the earth’s rotation. 
When the surfaces of the two bodies are almost in contact, the motion of the moon 
relatively to the earth’s surface would be almost insensible. This appears to point to 
the break-up of the primeval planet into two parts, in consequence of a rotation so 
rapid as to be inconsistent with an ellipsoidal form of equilibrium, 
Is it then a mere coincidence that the shortest period of revolution, with which a 
spheroid of the same mean density as the earth could subsist in the ellipsoidal form, 
is 2 hrs. 24 m. ; whilst if Kepler’s law were to hold true, and if the moon were to 
revolve round the earth in the same period, the surfaces of the two bodies would just 
graze one another ? 
§ 34. Summary of Parts V. and VI. 
I now come to the second of the two problems, where the moon moves in an 
eccentric orbit, always coincident with the ecliptic. 
In § 23 it is shown that the tides raised by any one satellite can produce no secular 
change in the eccentricity of the orbit of any other satellite; thus the eccentricity 
and the mean distance are in this respect on the same footing. 
It was found to be more convenient to consider the ellipticity of the orbit instead 
of the eccentricity. In § 24 (289) and (290), are given the time-rates of increase of 
the ellipticity and of the square root of mean distance. In § 25 the result for the 
ellipticity is applied to the case where the earth is viscous, and its physical meaning is 
graphically illustrated in fig. 8. 
This figure shows that in general the ellipticity will increase with the time; but if 
the obliquity of the ecliptic be nearly 90°, or if the viscosity be so great that the earth 
is very nearly rigid, the ellipticity will diminish. This last result is due to the rising 
into prominence of the effects of the elliptic monthly tide. 
If the viscosity be very small the equation is reducible to a very simple form, which 
is given in (291). From (291) we see that if the obliquity of the ecliptic be zero, the 
ellipticity will either increase or diminish, according as 18 rotations of the planet take 
a shorter or a longer time than 11 revolutions of the satellite. From this it follows 
that in the history of a satellite revolving about a planet of small viscosity, the circular 
orbit is dynamically stable until 11 months of the satellite have become longer than 
mdccclxxx. 5 u 
