4 
MR. DARWIN ON THE BODILY TIDES OF VISCOUS AND SEMI-ELASTIC 
— 7--f nV'~a+X = 0, &c., &e. 
n cL x 
m -3 
Also 
P= ——— -p-\-2n~, Q= &c., R = &c. 
n L ax 
m—5 
O 
Now if we suppose the elastic solid to he incompressible, so that m is infinitely 
large compared to n, then it is clear that the equations of equilibrium of the incom¬ 
pressible elastic solid assume exactly the same form as those of flow of the viscous 
fluid, n merely taking the place of v. 
Thus every problem in the equilibrium of an incompressible elastic solid has its 
counterpart in a problem touching the state of flow of an incompressible viscous fluid, 
when the effects of inertia are neglected ; and the solution of the one may be made 
applicable to the other by merely reading for “ displacements ” “ velocities,” and for 
the coefficient of “rigidity ” that of “ viscosity.” 
2. A sphere under influence of bodily force. 
Sir W. Thomson has solved the following problem :— 
To find the displacement of every point of the substance of an elastic sphere exposed 
to no surface traction, but deformed infinitesimally by an equilibrating system of forces 
acting bodily through the interior. 
If for “ displacement” we read velocity, and for “ elastic” viscous, we have the 
corresponding problem with respect to a viscous fluid, and mutatis mutandis the 
solution is the same. 
But we cannot find the tides of a viscous sphere by merely ipaking the equilibrating 
system of forces equal to the tide-generating influence of the sun or moon, because the 
substance of the sphere must be supposed to have the power of gravitation. 
For suppose that at any time the equation to the free -surface of the earth (as the 
co 
viscous sphere may be called for brevity) is r=a-j-^cr z -, where cr; is a surface harmonic. 
2 
Then the matter, positive or negative, filling the space represented by 'Icr, exercises 
an attraction on every point of the interior ; and this attraction, together with that of 
a homogeneous sphere of radius a, must be added to the tide-generating influence to 
form the whole force in the interior of the sphere. Also it is a spheroid, and no 
longer a true sphere with which we have to deal. If, however, we cut a true sphere 
of radius a out of the spheroid (leaving out -erf then by a proper choice of surface 
actions, the tidal problem may be reduced to finding the state of flow in a true sphere 
under the action of (i) an external tide-generating influence, (ii) the attraction of the 
