252 
THE TIDES. 
longitudes of the node and of the equinoctial line are not simply periodic in 
time, and may and will lead to cumulative effects. 
This explains what was stated above, namely, that we cannot entirely 
ignore the motion of the two nodes. 
The problem is thus divisible into three cases :— 
(i) Where the nodes revolve uniformly on the ecliptic, and where 
there is a second disturbing satellite, namely the sun. 
(ii) Where the earth and moon are the only bodies in existence. 
(iii) Where the nodes either oscillate, or do not revolve uniformly. 
The cases (i) and (ii) are distinguished by our being able to ignore the nodes. 
In none of the three cases can the tides generated by any one satellite 
produce directly a secular change in the mean distance of any other 
satellite. 
In cases (i) and (ii) the tides generated by any one satellite cannot 
produce any secular change in the inclination of the orbit of any other 
satellite to the plane of reference. This is not true of case (iii). 
Before proceeding further it may be well to remark that we shall have 
occasion to describe several of the elements as possessing stability or in 
stability. But these terms are not in strictness applicable, because, if any 
element is momentarily stable or unstable, the simultaneous changes in all 
the other elements will upset the permanence of that characteristic. The 
expression is however convenient and sufficiently explanatory. 
We will now consider case (i) where the moon’s node revolves uniformly 
on the ecliptic, and will state the results as to the rates of variation of the 
several elements. 
The obliquity of the ecliptic in general increases with the time. But if, 
with the month of its present length, the obliquity of the ecliptic were about 
87°, it would diminish. There is therefore a position of stability for a very 
large obliquity. The angle of obliquity which corresponds with the position 
of stability diminishes as the moon’s periodic time diminishes, and so long as 
the month is greater than twice the day zero obliquity is still a position of 
instability. But if the month is less than twice the day zero obliquity 
becomes stable. These results refer to the hypothesis that the amount of 
viscosity is small. 
Reverting to the case of the two disturbing bodies, it appears that the 
combined effect of sun and moon operates adversely to the separate effects 
of each. 
The tidal friction will always cause the earth’s diurnal rotation to 
diminish. The separate and combined effects cooperate, but the tides of 
long period have no effect. All the results which we are now stating depend