WITH THE TIDES OP A VISCOUS SPHEROID. 
545 
They are 
and 
«’a 3 -| - sin 4e sin 3 6 cos 0 sin 4 (<£ — &>#) along the meridian, 
wa ~-§■- sin 4e sin 3 0(1 + cos 4(<£ —wf)) perpendicular to the meridian. 
These stresses of course vanish when e is zero, that is to say when the spheroid 
is perfectly fluid. 
In as far as they involve <p— cot these expressions are periodic, and the periodic 
parts must correspond with periodic inequalities in the state of flow of the interior of 
the earth. These small tides of the second order have no present interest and 
may be neglected. 
We are left, therefore, with a non-periodic tangential stress per unit area of the 
surface of the sphere perpendicular to the meridian from east to west equal to 
\w($— sin 4e sin 3 0. 
5 
The sum of the moments of these stresses about the axis Z constitutes the tidal 
frictional couple Jl, which retards the earth’s rotation. 
Therefore 
jlt= | iva~- sin 4ej j sin 3 0. a sin 0.a~ sin 0d0cl 
integrated all over the surface of the sphere, and effecting the integration we have 
47T - T 2 
M = — wa° . — sm 4e. 
15 y 
But if C be the earth’s moment of inertia, C=^mva 5 . 
Therefore 
—sm 4e 
C 2 a 
. (5) 
This expression agrees with that found by a different method in the paper on 
“ Precession.”* 
We may now write the tangential stress on the surface of the sphere as ^wa~~ sin 3 0 ; 
L/ 
and the components of this stress parallel to the axes X, Y, Z are 
— lioar— sin 3 0 sin </>, -f sin 3 0 cos <f>, 0 
( 6 ) 
We now have to consider those effects of inertia which equilibrate this system of 
surface forces. 
The couple JT retards the earth’s rotation very nearly as though it were a rigid 
* “-Precession,” Section 5 (22), wlien 1=0. 
4 A 2 
