870 
MR, G. H. DARWIN ON THE SECULAR CHANGES IN 
The nodal velocity varies directly as the moon’s periodic time, and it will decrease 
as we look backwards in time. 
The processional velocity varies directly as the ellipticity of the earth’s figure (the 
earth being homogeneous) and inversely as the cube of the moon’s distance, and 
inversely as the earth’s diurnal rotation ; it will therefore increase retrospectively. 
The ratio of these two velocities is the quantity on which the position of the proper 
planes principally depends. 
The second cause of disturbance is due directly to the tidal interaction of the three 
bodies. 
The most prominent result of this interaction is, that the inclination of the lunar 
orbit to its proper plane in general diminishes as the time increases, or increases 
retrospectively. This statement may be compared with the results of Part II., where 
the ecliptic was in effect the proper plane. The retrospective increase of inclination 
may be reversed however, under special conditions of tidal disturbance and lunar 
periodic time. 
Also the inclination of the earth’s proper plane to the ecliptic in general increases 
with the time, or diminishes retrospectively. This is exemplified by the results of 
the paper on “Precession,” where the obliquity of the ecliptic was found to diminish 
retrospectively. This retrospective decrease may be reversed under special conditions. 
It is in determining the effects of this second set of causes, that we have to take 
account of the effects of tidal disturbance on the motions of the nodes of the orbit and 
equator on the ecliptic. 
After a long analytical investigation, equations are found in (224), which give the 
rate of change of the positions of the proper planes, and of the inclinations thereto. 
It is interesting to note how these equations degrade into those of case (i.) when 
the nodal velocity is very large compared with the precessional velocity, and into those 
of case (li.) when the same ratio is very small. 
In order completely to define the rate of change of the configuration of the system, 
there are two other equations, one of which gives the rate of increase of the square 
root of the moon’s distance (which I called in a previous paper the equation of tidal 
reaction), and the other-gives the rate of retardation of the earth’s diurnal rotation 
(which I called before the equation of tidal friction). For the latter of these we 
may however substitute another equation, in which the time is not involved, and 
which gives a relationship between the diurnal rotation and the square root of the 
moon’s distance. It is in fact the equation of conservation of moment of momentum 
of the moon-earth system, as modified by the solar tidal friction. This is the equa¬ 
tion which was extensively used in the paper on “ Precession.” 
Except for the solar tidal friction and for the obliquity of the orbit and equator, this 
equation would be rigorously independent of the kind of frictional tides existing in the 
earth. If the obliquities are taken as small, they do not enter in the equation, and in 
the present case the degree of viscosity of the earth only enters to an imperceptible 
