THE ELEMENTS OF THE ORBIT OF A SATELLITE. 
55 
If however X be not small, then even though the viscosity be not great enough to 
approach perfect rigidity, we must have sin 211*= 2(1—X) sin 4f u /X. And of course, by 
supposing the viscosity great enough, this relation may be fulfilled whatever be X. 
Then our equation becomes 
af 
T 2 h dt 
logp=-| 
. „..12 — 80A. + 96A, 3 —29A? 
mid xti _ 
(294) 
The numerator on the right-hand side is always positive for values of X less than 
unity, and the denominator is positive for values of X less than -§. 
Since 
we have 
ld_l 
k dt 
t.d , . 12 — 80X+ 96X 2 —29\ s 
lo «' *>= -I H 1-.X)(1-4X) 
From this we see that, for very large viscosity,— 
For values of X between 1 and '6667, the eccentricity increases per unit increase 
of £, and the rate of increase tends to become infinite when X='6667. 
The remarks concerning the physical absurdity of this class of result in § 21 may be 
repeated in this case. 
And for values of X between '6667 and 0, the eccentricity diminishes. 
A similar treatment of the case of small viscosity shows that— 
For values of X between 1 and '6111 the eccentricity decreases, and for values of X 
between '6111 and 0 the eccentricity increases. 
Thus it is only between X= *6111 and "6667 that the two cases agree. 
Hence in the course of evolution of a satellite revolving about a purely viscous 
planet:— 
For small viscosity the orbit will remain circular until 11 months of the satellite are 
equal to 18 days of the planet, then the eccentricity will increase until this relation¬ 
ship is again fulfilled, when the eccentricity will again diminish.* 
And for very large viscosity the orbit will at once become eccentric, and the 
eccentricity will increase very rapidly until two months of the satellite are equal to 
three days of the planet. The eccentricity will then diminish until this relationship 
is again fulfilled, after which the eccentricity will again increase. 
We shall consider later which of these views seems the more probable with regard 
to the history of the moon. 
* See “ On the Analytical Expressions, &c.,” Proc. Roy. Soc., No. 202, 1880. 
5 R 2 
