WITH THE TIDES OF A VISCOUS SPHEROID. 
549 
w /) 
w 
& 
sin 3 6 sin </>, fi= - -ffr 3 sin 3 6 cos <f>, y=0 
ov U 
(17) 
These values show that the motion is simply cylindrical round the earth’s axis, each 
point moving in a small circle of latitude from east to west with a linear velocity 
~ ~7' 3 sin 3 0, or with an angular velocity 
In this statement, a meridian at the pole is the curve of reference, but it is more 
intelligible to state that each particle moves from west to east with an angular velocity 
about the axis equal to ~ yr(a 2 —r 2 sin 2 6), with reference to a point on the surface at 
the equator. 
The easterly rate of change of the longitude L of any point on the surface in 
colatitude 6 is therefore ~ yy cos 2 6. 
ov 0 
about the axis equal to — —V 2 sin 2 9 .* 
Then since 
§ =- sin 2e cos 2e, and tan 2<=f 
C g ’ 5 
19t iu> 
g wa 2 ’ 
therefore 
^_JL9 
dt ~ 20 
- cos 2e 
g 
a) cos 2 6 
(17') 
This equation gives the rate of change of longitude. The solution is not applicable 
to the case of perfect fluidity, because the terms introduced by inertia in the equations 
of motion have been neglected; and if the viscosity be infinitely small, the inertia 
terms are no longer small compared wflth those introduced by viscosity. 
In order to find the total change of longitude in a given period, it will be more 
convenient to proceed from a different formula. 
Let n, fl be the earth’s rotation, and the moon’s orbital motion at any time ; and let 
the suffix 0 to any symbol denote its initial value, also let d= 
Then it was shown in the paper on “Precession” that the equation of conservation 
of moment of momentum of the moon-earth system is 
( 18 ) 
Where /x is a certain constant, which in the case of the homogeneous earth with the 
present lengths of day and month, is almost exactly equal to 4. 
By differentiation of (18) 
chi 
* The problem might probably be solved more shortly without using the general solution, but the 
general solution will be required in Part III. 
f “Precession,” equation (73), when i =0 and T = 0. 
