448 
MR. G. H. DARWIN ON THE PRECESSION OF A VISCOUS SPHEROID, 
As some persons may wish to obtain a general idea of the drift of the inquiry with¬ 
out reading a long mathematical argument, I have adhered to the plan adopted in my 
former paper, of giving at the end (in Part III.) a general view of the whole subject, 
with references back to such parts as it did not seem desirable to reproduce. In order 
not to interrupt the mathematical argument in the body of the paper, the discussion of 
the physical significance of the several results is given along with the summary ; such 
discussions will moreover be far more satisfactory when thrown into a continuous 
form than when scattered in isolated paragraphs throughout the paper. I have tried, 
however, to prevent the mathematical part from being too bald of comments, and to 
place the reader in a position to comprehend the general line of investigation. 
Before entering on analysis, it is necessary to give an explanation of how this 
inquiry joins itself on to that of my previous paper. 
In that paper it was shown that, if the influence of the disturbing body be expressed 
in the form of a potential, and if that potential be expressed as a series of solid 
harmonic functions of points within the disturbed spheroid, each multiplied by a simple 
time harmonic, then each such harmonic term raises a tide in the disturbed spheroid, 
which is the same as though all the other terms were non-existent. This is true, 
whether the spheroid be fluid, elastic, viscous, or elastico-viscous. Further, the free 
surface of the spheroid, as tidally distorted by any term, is expressible by a surface 
harmonic of the same type as that of the generating term; and where there is a 
frictional resistance to the tidal motion, the phase of the corresponding simple time 
harmonic is retarded. The height of each tide, and the retardation of phase (or the 
lag) are functions of the frequency of the tide, and of the constants expressive of the 
physical constitution of the spheroid. 
Each such term in the expression for the form of the tidally distorted spheroid may 
be conveniently referred to as a simple tide. 
Hence if we regard the whole tide-wave as a modification of the equilibrium tide- 
wave of a perfectly fluid spheroid, it may be said that the effect of the resistances to 
relative displacement is a disintegration of the whole wave into its constituent simple 
tides, each of which is reduced in height, and lags in time by its own special amount. 
In fact, the mathematical expansion in surface harmonics exactly corresponds to the 
physical breaking up of a single wave into a number of secondary waves. 
It was remarked in the previous paper,* that when the tide-wave lags the attraction 
of the external tide-generating body gives rise to forces on the spheroid which are not 
rigorously equilibrating. Now it was a part of the assumptions, under which the 
theory of viscous and elastico-viscous tides was formed, that the whole forces which 
act on the spheroid should be equilibrating ; but it was there stated that the couples 
arising from the non-equilibration of the attractions on the lagging tides 'were pro¬ 
portional to the square of the disturbing influence, and it was on this account that 
they were neglected in forming that theory of tides. The investigation of the effects 
* “ Bodily Tides,” &c. Sec. 5. 
