540 
MR. G. H. DARWIN ON PROBLEMS CONNECTED 
the mean sphere—or more strictly the mean spheroid of revolution which represents 
the average shape of the earth. The spheroid was endued with the power of gravi¬ 
tation, and it was shown that the action of the spheroid on its own tides might be 
found approximately by considering the state of flow in the mean sphere caused by 
the attraction of the protuberances, and also by supposing the action of the protu¬ 
berances on the sphere to be normal thereto, and to consist, in fact, merely of the 
weight (either positive or negative) of the protuberances. 
Thus if a be the mean radius of the sphere, w its density, g mean gravity at the 
surface, and r=a-j-cr; the equation to the tidal protuberance, where cr; is a surface 
harmonic of order i, the potential per unit volume of the protuberance in the interior 
of the sphere is - ) cr/, and the sphere is subjected to a normal traction per unit 
area of surface equal to — giver /. 
It was also shown that these two actions might be compounded by considering the 
interior of the sphere (now free of gravitation) to be under the action of a potential 
2(i-l) 
2i+lS W {a)^ 
This expression therefore gave the effective potential when the sphere was treated 
as devoid of gravitational power. 
It was remarked* that, strictly speaking, there is tangential action between the pro¬ 
tuberance and the surface of the sphere. And latert it was stated that the action of 
an external tide-generating body on the lagging tides was not such as to form a 
rigorously equilibrating system of forces. The effects of this non-equilibration, in as 
far as it modifies the rotation of the spheroid as a whole, were considered in the paper 
on “Precession.” 
It is easy to see from general considerations that these previously neglected tangential 
stresses on the surface of the sphere, together with the effects of inertia due to the 
secular retardation of the earth’s rotation (produced by the non-equilibrating forces), 
must cause a secular distortion of the spheroid. 
This distortion I now propose to investigate. 
In order to avoid unnecessary complication, the tides will be supposed to be raised 
by a single disturbing body or moon moving in the plane of the earth’s equator. 
Let r=o-\-ar be the equation to the bounding surface of the tidafly-distorted earth, 
where cr is a surface harmonic of the second order. 
I shall now consider how the equilibrium is maintained of the layer of matter cr, as 
acted on by the attraction of the spheroid and under the influence of an external dis¬ 
turbing potential V, which is a solid harmonic of the second degree of the coordinates 
of points within the sphere.| The object to be attained is the evaluation of the stresses 
# “ Tides,” Section 2. 
f “ Tides,” Section 5. 
| A parallel investigation would be applicable, where a and V are of any orders. 
