526 MR. G. H. DARWIN ON THE PRECESSION OF A VISCOUS SPHEROID 
replaced by two circular rings of matter coincident with their orbits and equal in mass 
to them respectively. The tidal friction due to these rings is insignificant compared 
with that arising separately from the sun and moon. But the diurnal combined effect 
has an important influence in affecting the rate of change of obliquity. The combined 
effects are such as to cause the obliquity of the ecliptic to diminish, whereas the 
separate effects on the whole make it increase—at least in general (see Section 22). 
The relative importance of all the effects may be seen from an inspection of Table III., 
Section 15. 
Section 11 contains a graphical analysis of the physical meaning of the equations, 
giving the rate of change of obliquity for various degrees of viscosity and obliquity. 
Plate 36, figures 2 and 3, refer to the case where the disturbed planet is the earth, 
and the disturbing bodies the sun and moon. 
This analysis gives some remarkable results as to the dynamical stability or 
instability of the system. 
It will be here sufficient to state that, for moderate degrees of viscosity, the position 
of zero obliquity is unstable, but that there is a position of stability at a high obliquity. 
For large viscosities the position of zero obliquity becomes stable, and (except for a 
very close approximation to rigidity) there is an unstable position at a larger obliquity, 
and again a stable one at a still larger one."' 
These positions of dynamical equilibrium do not rigorously deserve the name, since 
they are slowly shifting in consequence of the effects of tidal friction ; they are rather 
positions in which the rate of change of obliquity becomes of a higher order of small 
quantities. 
It appears that the degree of viscosity of the earth which at the present time would 
cause the obliquity of the ecliptic to increase most rapidly is such that the bodily semi¬ 
diurnal tide would be retarded by about 1 hour and 10 minutes ; and the viscosity 
which would cause the obliquity to decrease most rapidly is such that the bodily semi¬ 
diurnal tide would be retarded by about 2f hours. 
The former of these two viscosities was the one which I chose for subsequent 
numerical application, and for the consideration of secular changes in the system. 
Plate 36, fig. 4 (Section 11), shows a similar analysis of the case where there is only 
one disturbing satellite, which moves orbitally with one-fifth of the velocity of rotation 
of the planet. This case differs from the preceding one in the fact that the position of 
zero obliquity is now unstable for all viscosities, and that there is always one other, 
and only one other position of equilibrium, and that is a stable one. 
This shows that the fact that the earth's obliquity would diminish for large viscosity 
is due to the attraction of the sun on the lunar tides, and of the moon on the solar 
tides. 
It is not shown by these figures, but it is the fact that if the motion of the satellite 
* For a general explanation of some part of. these results, see the abstract of this paper in the 
4 Proceedings of the Royal Society,’ No. 191,1878. 
