THE ELEMENTS OE THE ORBIT OF A SATELLITE. 
889 
henceforth suppose that h is greater than 4/3*. It will be seen presently that in this 
case every section parallel to x has a maximum and minimum point, and the nature 
of the sections is exhibited in the curves of energy in the two previous papers. 
Now consider the condition n=S2, which expresses that the planet rotates in the 
same period as that in which the satellite revolves, so that if the orbit be circular the 
two bodies revolve like a single rigid body. 
With the present units f2= 1 /as 3 , and by (a), n=h—x(l—rj). 
Hence the condition n=f2 leads to the biquadratic 
x * 
-x 
1—7] 1—7] 
= 0 
(S) 
If r] be zero this equation is identical with (y), which gives the maxima and minima 
of energy. 
Hence if the orbit be circular the maximum and minimum of energy correspond to 
two cases in which the system moves as a rigid body. If however the orbit be 
elliptical, and if n=f2, there is still relative motion during revolution of the satellite, 
and the energy must be capable of degradation. The principal object of the present 
note is to investigate the stability of the circular orbit in these cases, and this question 
involves a determination of the nature of the degradation when the orbit is elliptical. 
In Part V. of the present paper it has been shown that if the planet be a fluid of small 
viscosity the ellipticity of the satellite’s orbit will increase if 18 rotations of the planet 
be less than 11 revolutions of the satellite, and vice versd. Hence the critical relation 
between n and fl is This leads to the biquadratic 
h ,,18 1 
.x 4 — x 3 +— = 0 
1 — 7] 11 1 — 7] 
This is an equation with two real roots, and when it is illustrated graphically it will 
lead to a pair of curves. For configurations of the system represented by points lying 
between these curves the eccentricity increases, and outside it diminishes,—supposing 
the viscosity of the planet and the eccentricity of the satellite’s orbit to be small. 
In order to illustrate the surface of energy (/3) and the three biquadratics (y), (S), 
and (c), I chose h= 3, which is greater than 4/3*. 
By means of a series of solutions, for several values of rj, of the equations (y), (S), (e), 
and a method of graphical interpolation, I have drawn the accompanying figure. 
The horizontal axis is that of x, the square root of the satellite’s distance, and the 
numbers written along it are the various values of x. The vertical axis is that of r), 
and it comprises values of rj between 0 and 1. The axis of z is perpendicular to the 
plane of the paper, but the contour lines for various values of 2 are projected on to the 
plane of the paper, 
