524 
MR. G. H. DARWIN ON THE PRECESSION OF A VISCOUS SPHEROID, 
earth’s axis. The case becomes enormously more complex if we suppose the moon to 
move in an inclined eccentric orbit with revolving nodes. The consideration of the 
secular changes in the inclination of the lunar orbit and of the eccentricity will form 
the subject of another investigation. 
The expression for the moon’s tide-generating potential is shown to consist of 13 
simple tide-generating terms, and the physical meaning of this expansion is given in 
the note to Section 8. The physical causes represented by these 13 terms raise 13 simple 
tides in the earth, the heights and retardations of which depend on their speeds and on 
the coefficient of viscosity. 
The 13 simple tides may be more easily represented both physically and analytically 
as seven tides, of which three are approximately semi-diurnal, three approximately 
diurnal, and one has a period equal to a half of the sidereal month, and is therefore 
called the fortnightly tide. 
Then by an approximation which is sufficiently exact for a great part of the investi¬ 
gation, the semi-diurnal tides may be grouped together, and the diurnal ones also. 
Hence the earth may be regarded as distorted by two complex tides, namely, the semi¬ 
diurnal and diurnal, and one simple tide, namely, the fortnightly. The absolute heights 
and retardations of these three tides are expressed by six functions of then* speeds and 
of the coefficient of viscosity (Sections 1 and 2). 
When the form of the distorted spheroid is thus given, the couples about three axes 
fixed in the earth due to the attraction of the moon on the tidal protuberances are 
found. It must here be remarked that this attraction must in reality cause a tan¬ 
gential stress between the tidal protuberances and the true surface of the mean 
oblate spheroid. This tangential stress must cause a certain very small tangential 
flow,'" and hence must ensue a very small diminution of the couples. The diminution 
of couple is here neglected, and the tidal spheroid is regarded as being instantaneously 
rigidly connected with the rotating spheroid. The full expression for the couples on 
the earth are long and complex, but since the nutations to which they give rise are 
exceedingly minute, they may be much abridged by the omission of all terms except 
such as can give rise to secular changes in the precession, the obliquity of the ecliptic, 
and the diurnal rotation. The terms retained represent that there are three couples 
independent of the time, the first of which tends to make the earth rotate about an 
axis in the equator which is always 90° from the nodes of the moon’s orbit: this 
couple affects the obliquity to the ecliptic ; second, there is a couple about an axis in 
1 1 fT elf, 
periodic time of the moon, the average value of y is qs ( If c t> e the mean distance and e the eccen- 
p p _|_3 e 2_|_3. e 4 
tricity of the orbit, this integi’al will be found equal to ^ ^ the eccen ! r ^ c ffi r be small the 
average value of ^ is q ^1 + qye 2 ^; if e is ^ this is ^ of ~ 6 . There are obviously forces tending to 
modify the eccentricity of the moon's orbit. 
* See Part I. of the next paper. 
