AND ON THE REMOTE HISTORY OF THE EARTH. 
449 
which they produce in modifying the relative motion of the parts of the spheroid, that 
is to say in distorting the spheroid, must be reserved for a future occasion.'" 
The effects of these couples, in modifying the motion of the rotating spheroid as a 
whole, affords the subject, of the present paper. 
According to the ordinary theory, the tide-generating potential of the disturbing 
body is expressible as a series of Legendre’s coefficients; the term of the first order 
is non-existent, and the one of the second order has the type fcos 2 — Throughout 
this paper the potential is treated as though the term of the second order existed alone, 
but at the end it is shown that the term of the third order (of the type f-cos 3 — f cos) 
will have an effect which is fairly negligeable compared with that of the first term. 
In order to apply the theory of elastic, viscous, and elastico-viscous tides, the first 
task is to express the tide-generating potential in the form of a series of solid harmonics 
relatively to axes fixed in the spheroid, each harmonic being multiplied by a simple 
time harmonic. 
Afterwards it wbll be necessary to express that the wave surface of the distorted 
spheroid is the disintegration into simple lagging tides of the equilibrium tide-wave of 
a perfectly fluid spheroid. 
The symbols expressive of the disintegration and lagging will be kept perfectly 
general, so that the theory will be applicable either to the assumptions of elasticity, 
viscosity, or elastico-viscosity, and probably to any other continuous law of resistance 
to relative displacement. It would not, however, be applicable to such a law as that 
which is supposed to govern the resistance to slipping of loose earth, nor to any law 
which assumes that there is no relative displacement of the parts of the solid, until 
the stresses have reached a definite magnitude. 
After the form of the distorted spheroid has been found, the couples which arise 
from the attraction of the disturbing body on the wave surface will be found, and the 
rotation of the spheroid and the reaction on the disturbing body will be considered. 
This preliminary explanation will, I think, make sufficiently clear the objects of the 
rather long introductory investigations which are necessary. 
PART I. 
§ 1. The tide-generating potent ial. 
The disturbing body, or moon, is supposed to move in a circular orbit, with a 
uniform angular velocity —fl. The plane of the orbit is that of the ecliptic ; for the 
investigation is sufficiently involved without complicating it by giving the true 
inclined eccentric orbit, with revolving nodes. [I hope however in a future paper to 
consider the secular changes in the inclination and eccentricity of the orbit and the 
modifications to be made in the results of the present investigation.] 
See tlie next paper “ On Problems connected with the Tides of a Viscous Spheroid.” Part I. 
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