543 
WITH THE TIDES OF A VISCOUS SPHEROID. 
Now these are the forces on the element which must be balanced by the tangential 
stresses across the base of the prismatic element. 
It follows from the above formulas that the tangential stresses communicated by the 
layer cr to the surface of the sphere are those due to a potential V — gcnr acting on 
the layer cr. 
V 
If cr=:— there is no tangential stress. But this is the condition that cr should be 
m & 
the equilibrium tidal spheroid due to Y, so that the result fulfils .the condition that if 
cr be the equilibrium tidal spheroid of Y there is no tendency to distort the spheroid 
further; this obviously ought to be the case. 
In the problem before us, however, cr does not fulfil this condition, and therefore 
there is tangential stress across the base of each prismatic element tending to distort 
the sphere. 
Suppose Y=r~S where S is a surface harmonic. 
Then at the surface V=orS. If Sm be the mass of a prism cut out of the layer cr, 
which stands on unit area as base, then Sm= wcr. 
Therefore the tangential stresses per unit area communicated to the sphere are 
and 
iva,"- 
(S — fly) along the meridian 
wa 2 --r—z~ (S — fly) perpendicular to the meridian 
a sm 9 clip v "br 1 1 
Besides these tangential stresses there is a small radial stress over and above the 
radial traction— giver, which was taken into account in forming the tidal theory. But 
we remark that the part of this stress, which is periodic in time, will cause a very 
small tide of the second order, and the part which is non-periodic will cause a very 
small permanent modification of the figure of the sphere. But these effects are so 
minute as not to be worth investigating. 
We will now apply these results to the tidal problem. 
4 A 
MDCCCLXXIX. 
