AND ON THE REMOTE HISTORY OF THE EARTH. 
4G5 
§ 8. The disturbing action of the sun , 
Now suppose that there is a second disturbing body, which may be conveniently 
called the sun.* 
* It is not at first sight obvious bow it is physically possible that the sun should exercise an influence 
on the moon-tide, and the moon on the sun-tide, so as to produce a secular change in the obliquity of the 
ecliptic and to cause tidal friction, for the periods of the sun and moon about the earth are different. It 
seems, therefore, interesting to give a physical meaning to the expansion of the tide-generating potential; 
it will then be seen that the interaction with which we are here dealing must occur. 
The expansion of the potential given in Section 1 is equivalent to the following statement: — 
The tide-generating potential of a moon of mass to, moving in a circular orbit of obliquity j at a 
distance c, is equal to the tide-generating potential of ten satellites at the same distance, whose orbits, 
masses, and angular velocities are as follows :— 
1. A satellite of mass m cos 4 moving in the equator in the same direction and with the same angular 
Li 
velocity as the moon, and coincident with it at the nodes. This gives the slow semi-diurnal tide of 
speed 2(n— Q). 
2. A satellite of mass to sin 4 moving in the equator in the opposite direction from that of the moon, 
but with the same angular velocity, and coincident with it at the nodes. This gives the fast semi-diurnal 
tide of speed 2(u + O). 
3. A satellite of mass m2 sin 2 cos 2 ~, fixed at the moon’s node. This gives the sidereal semi-diurnal 
Li —' 
tide of speed 2 n. 
4. A repulsive satellite of mass — to. 2 sin 2’ cos 3 moving in N. declination 45 D with twice the moon’s 
angular velocity, in the same direction as the moon, and on the colure 90° in advance of the moon, when 
she is in her node. 
5. A satellite of mass to sin % cos 3 —, moving in the equator with twice the moon’s angular velocity, and 
Li 
in the same direction, and always on the same meridian as the fourth satellite. (4) and (5) give the slow 
diurnal tide of speed n— 20. 
6. A satellite of mass to sin 3 ^ cosb, moving in N. declination 45° with twice the moon’s angular velocity, 
Li Li 
but in the opposite direction, and on the colure 90° in advance of the moon when she is in her node. 
7. A repulsive satellite of mass —to. ^ sin 3 ^cos-^ , moving in the equator with twice the moon’s angular 
Li Li Li 
velocity, but in the opposite direction, and always on the same meridian as the sixth satellite. (6) and 
(7) give the fast semi-diurnal tide of n + 2D. 
8. A satellite of mass to sin i cos i fixed in N. declination 45° on the colure. 
9. A repulsive satellite of mass —to. i sin i cos i, fixed in the equator on the same meridian as the eighth 
Li 
satellite. (8) and (9) give the sidereal diurnal tide of speed n. 
10. A ring of matter of mass to, always passing through the moon and always parallel to the equator. 
This ring, of course, executes a simple harmonic motion in declination, and its mean position is the 
equator. This gives the fortnightly tide of speed 2D. 
Now if we form the potentials of each of these satellites, and omit those parts which, being indepen¬ 
dent of the time, are incapable of raising tides, and add them altogether, we shall obtain the expansion 
for the moon’s tide-generating potential used above ; hence this system of satellites is mechanically 
3 o 2 
