THE ELEMENTS OP THE ORBIT OP A SATELLITE. 
835 
or 
a:3 = ,i »+i-«; +K 
Uo+7+Kf+-=0 
The root of this equation, which gives the required phase, is nearly equal to the 
cube-root of the second coefficient, hence 
n 0 -f- + K ) nearly. 
Now in the paper on “Precession” we found the initial condition, on the hypothesis 
that K was zero. Hence the effect of solar tidal friction is to increase the angular 
velocity of the two bodies when their relative motion is zero. Since K may be large, 
it follows that the disturbance of the solution of § 16 of “Precession” may be 
considerable. 
This therefore shows that it is probable that an accurate solution of our problem 
would differ considerably from that found in “Precession,” and that the common 
angular velocity of the two bodies might have been very great. 
If Kepler’s law holds good, then the periodic time of the moon about the earth, 
when their centres are 6,000 miles apart, is 2 hrs. 36 m., and when 5,000 miles apart 
is 1 hr. 57 m. ; hence when the two spheroids are just in contact, the time of 
revolution of the moon would be between 2 hrs. and 2^ hrs. 
Now it is a remarkable fact that the most rapid rate of revolution of a mass of fluid, 
of the same mean density as the earth, which is consistent with an ellipsoidal form of 
equilibrium, is 2 hrs. 24 m. Is this a mere coincidence, or does it not rather point 
to the break-up of the primaeval planet into two masses in consequence of a too rapid 
rotation ? 
*Tt is not possible to make an adequate consideration of the subject of this section 
without a treatment of the theory of the tidal friction of a planet attended by a pair 
of satellites. 
It was shown above that if the moon were to move orbitally nearly as fast as the 
earth rotates, the solar tidal friction would be more important than the lunar, however 
near the moon might be to the earth. I now (September, 1880) find that the con¬ 
sequence of this is that the earth’s rotation continues to increase retrospectively, and 
the moon’s orbital motion does the same ; but the difference of the rotation and orbital 
* From this point to the end has been added, and the section otherwise abridged since the paper was 
presented.—September, 1880. 
