SPHEROIDS, AND ON THE OCEAN TIDES UPON A YIELDING NUCLEUS. 29 
here assumed lias not, as far as I am aware, any experimental justification; its 
adoption was rather due to mathematical necessities than to any other reason. 
It would, of course, have been much more interesting if it had been possible to 
represent more exactly the mechanical properties of solid matter. One of the most 
important of these is that form of resistance to relative displacement, to which the 
term “ plasticity ” has been specially appropriated. This form of resistance is such that 
there is a change in the law of resistance to the relative motion of the parts, when the 
forces tending to cause flow have reached a certain definite intensity. This idea was 
founded, I believe, by MM. Tresca and St. Venant on a long course of experiments 
on the punching and squeezing of metals and they speak of a solid being reduced to 
the state of fluidity by stresses of a given magnitude. This theory introduces a 
discontinuity, since it has to be determined what parts of the body are reduced to the 
state of fluidity and what are not. But apart from this difficulty, there is another 
one which is almost insuperable, in the fact that the differential equations of flow are 
non-linear. 
The hope of introducing this form of resistance must be abandoned, and the investi¬ 
gation must be confined to the inclusion of those two other continuous laws of resistance 
to relative displacement—elasticity and viscosity. 
As above stated, the law of elastico-viscosity assumed in this paper has not got an 
experimental foundation. Indeed, Kohlrausch’s experiments on glass! show that 
the elasticity degrades rapidly at first, and that it tends to attain a final condition, 
from which it does not seem to vary for an almost indefinite time. But glass is one 
of the most perfectly elastic substances known, and, by the light of Tresca’s experi¬ 
ments, it seems probable that experiments with lead would have brought out very 
different results. It seems, moreover, hardly reasonable to suppose that the materials 
of the earth possess much mechanical similarity with glass. Notwithstanding all 
these objections, I think, for my part, that the results of this investigation of the tides 
of an ideal elastico-viscous sphere are worthy of attention. 
There are two constants which determine the nature of this ideal solid first, the 
coefficient of rigidity, at the instant immediately after the body has been placed in its 
strained configuration; and secondly, “ the modulus of the time of relaxation ol 
rigidity,” which is the time in which the force requisite to retain the body in its 
strained configuration has fallen away to ‘368 of its initial value. 
In this section it is shown that the equations of flow of this incompressible elastico- 
viscous body have the same mathematical form as those for a purely viscous body; so 
that the solutions already attained are easily adapted to the new hypothesis. 
The only case where the problem is completely worked out, is when the disturbing 
* “ Sur 1’ecoulement des Corps Solides,” Mem. des Savants Etrangers, tom. xviii. and tom. xx., p. 75 
and p. 137. See also ‘ Comptes Rendus,’ tom. 66, 68, and Liouville’s Jonrn., 2 me serie, xiii., p. 379, and 
xvi., p. 308, for papers on this subject. 
| Poggendorff Ann., vol. 119, p. 337. 
