22 
MR. DARWIN ON THE BODILY TIDES OE VISCOUS AND SEMI-ELASTIC 
and this function gives the proposed constrained oscillation. It is clear that it is the 
particles at the surface which have the widest amplitude of oscillation. The effective 
force on a unit element at the surface is 
oa 
—w -p = — wv 2 S cos (yt-\-rj). 
But the normal disturbing force at the surface is 2wa S cos Therefore inertia 
5a 2 . 5a 
may be neglected if — ivv 2 is small compared with 2 wa, or if — v' 2 is a small fraction. 
cJ 
The tide of the shortest period with which we have to deal is that in which v=2oj, so that 
5am 2 T 
It co were the earths true 
we must consider the magnitude of the fraction 4 X , . 
o ctco^ 
angular velocity, instead of its angular velocity relatively to the moon, then —— would 
be the ellipticity of its surface if it were homogeneous. This ellipticity is, as is well 
known, Hence the fraction, which is the criterion of the negligeability of inertia, 
is about - 5 - 8 -. 
If, then, it be considered that this way of looking at the subject certainly exag¬ 
gerates the influence of inertia, it is clear that the neglect of inertia is not such as to 
materially vitiate the results given above. 
II. 
A TIDAL YIELDING OP THE EARTH’S MASS, AND THE CANAL-THEORY 
OF TIDES. 
In the first part of this paper the equilibrium theory has been used for the determi¬ 
nation of the reduction of the height of tide, and the alteration of phase, due to bodily 
tides in the earth. 
Sir W. Thomson remarks, with reference to a supposed elastic yielding of the 
earth’s body : “ Imperfect as the comparisons between theory and observation as to 
the actual height of the tides has been hitherto, it is scarcely possible to believe that 
the height is in reality only two-fifths of what it would be if, as has been universally 
assumed in tidal theories, the earth were perfectly rigid. It seems, therefore, nearly 
certain, with no other evidence than is afforded by the tides, that the tidal effective 
rigidity of the earth must be greater than that of glass.”* 
The equilibrium theory is quite fallacious in its explanation of the semidiurnal tide, 
but Sir W. Thomson is of opinion that it must give approximately correct results 
for tides of considerable period. It is therefore on the observed amount of the 
fortnightly tide that he places reliance in drawing the above conclusion. Under these 
* ‘ Nat. Phil,’, § 843. 
