878 
MR. G. H. DARWIN ON THE SECULAR CHANGES IN 
18 days of the planet. Since the day and month start from equality and end in 
equality, it follows that the eccentricity will rise to a maximum and ultimately 
diminish again. 
It is also shown that if a satellite be started to move in a circular orbit with the 
same periodic time as that of the planet’s rotation (with maximum energy for given 
moment of momentum), then if infinitesimal eccentricity be given to the orbit the 
satellite will ultimately fall into the planet; and if, the orbit being circular, infini¬ 
tesimal decrease of distance be given the satellite will fall in, whilst if infinitesimal 
increase of distance be given the satellite will recede from the planet. Thus this con¬ 
figuration, in which the planet and satellite move as parts of a single rigid body, has a 
complex instability; for there are two sorts of disturbance which cause the satellite 
to fall iu, and one which causes it to recede from the planet.'"' 
If the planet have very large viscosity the case is much more complex, and it is 
examined in detail in § 25. 
It will here only be stated that the eccentricity will diminish if 2 months of the 
satellite be longer than 3 days of the planet, but will increase if the 2 months be 
shorter than 3 days ; also the rate of increase of eccentricity tends to become infinite, 
for infinitely great viscosity, if the 2 months are equal to the 3 days. 
These results are largely due to the influence of the elliptic monthly tide, and with 
most of the satellites of the solar system, this is a very slow tide compared with the 
semi-diurnal tides; therefore it must in general be supposed that the viscosity of the 
planet makes a close approximation to perfect rigidity, in order that this statement 
may be true. 
The infinite value of the rate of change of eccentricity is due to the speed of the 
slower elliptic semi-diurnal tide being infinitely slow, when 2 months are equal to 
3 days. The result is physically absurd, and its true meaning is commented on 
in § 25. 
In § 26 the time-rate of change of the obliquity of the planet’s equator, and of the 
diurnal rotation is investigated, when the orbits of the tide-raising satellites are 
eccentric ; the only point of general interest in the result is, that the rate of change of 
obliquity and the tidal friction are both augmented by the eccentricity of the orbit, 
as was foreseen in the paper on “ Precession.” 
In § 27 it is stated that the effect of the evectional tides is such as to diminish the 
eccentricity of the orbit, but the formula given shows that the effect cannot have 
much importance, unless the moon be very distant from the earth. 
* Added July, 1880.—This passage appeared to the referee, requested by the R. S. to report on this 
paper, to be rather obscure, and it has therefore been somewhat modified. To further elucidate the point 
I have added in an appendix a graphical illustration of the effects of eccentricity, similar to those given 
in No. 197 of Proc. Eoy. Soc., 1879. 
See also the abstract of this paper in the Proc. Roy. Soc., No. 200, 1879, for certain general con¬ 
siderations bearing on the problem of the eccentricity. 
