6 
MR. DARWIN ON THE BODILY TIDES OF VISCOUS AND SEMI-ELASTIC 
with symmetrical expressions for /3 and y; where T r and are auxiliary functions 
defined by 
*<-! = 
®i+i= 
:±(AS)+%(Bri+£( 
’• Si+3 {|( A ' ! - M ) + |(B,r- 1 ) + |(C, -- 1 ) 
1 
h (6) 
In the case considered by Sir W. Thomson of an elastic sphere deformed by bodily 
stress and subject to no surface action, we have to substitute ill (5) and (6) only those 
surface actions which are equal and opposite to the surface forces corresponding to the 
first part of the solution ;* but in the case which we now wish to consider, we must 
add to these latter the components of the normal traction — givScr,, and besides must 
include in the bodily force both the external disturbing force, and the attraction of 
the matter of the spheroid on itself. 
Now from the forms of (5) and (6) it is obvious that the tractions which correspond 
to the first part of the solution, and the traction — gw%cr l produce quite independent 
effects, and therefore we need only add to the complete solution of Sir W. Thomson’s 
problem of the elastic sphere, the terms which arise from the normal traction — gw%cri. 
Finally we must pass from the elastic problem to the viscous one, by reading v for n, 
and velocities for displacements. 
I proceed then to find the state of internal flow in the viscous sphere, which results 
from a normal traction at every point of the surface of the sphere, given by the 
surface harmonic S,-. 
In order to use the formulae (5) and (6), it is first necessary to express the 
component tractions - S;, y S/, - S; as surface harmonics. 
Now if Yi be a solid harmonic. 
So that 
Therefore 
! '- l V;) - (2i+ l)r-tt‘ +S) xYi+ 
dVi 
dx 
The quantities within the brackets [ ] being independent of r, and being 
X 
surface harmonics of orders i — 1 and i +1 respectively, we have - S; expressed as the 
sum of two surface harmonics A;_ 1? A, +1 , where 
* Where the solid is incompressible, this surface traction is normal to the sphere at every point, 
provided that the potential of the bodily force is expressible in a series of solid harmonics. 
