200 
MR. G-. H. DARWIN ON THE STRESSES 
§ 4. The application of previous analysis to the determination of the stresses 
produced hy the weight of superficial inequalities. 
I have in a previous paper shown how Sir William Thomson’s solution for the 
state of internal strain of an elastic sphere subject to bodily forces, but not acted on 
by any surface forces, is to be adapted to the case of a spheroid (whose small 
inequalities are expressed as surface harmonics) of homogeneous elastic matter, endued 
with the power of mutual gravitation.* Thomson’s solution is of course directly 
applicable for finding the state of strain due to a true external force, such as the tide¬ 
generating influence of the moon, but this forms only a part of the complete solution 
when the sphere has the power of gravitation. He introduced the effects of gravita¬ 
tion synthetically, but for my own purposes I prefer the analytical method pursued in 
my paper above referred to. 
Suppose that r=a+ov be the equation to an harmonic spheroid of the i th order, 
forming inequalities on the surface of the sphere, whose density is w. 
Then the causes producing a state of stress and strain in the mean sphere of radius 
a are, first a normal traction per unit area of the surface of the sphere equal to — gwcn , 
when g is the value of gravity, and secondly the attraction of the inequalities o’,-, acting 
throughout the whole sphere. 
The first of these causes (viz. : the weight of the mountains or continents) is shown 
in my paper to produce the same state of strain as would be produced in the sphere, 
now free from surface action, by bodily forces corresponding with a potential 
—giu(r/a) ; (Ti. 
As regards the second of these causes (viz. : the attraction of the mountains or 
continents), the potential of the layer of matter <n on any internal point, estimated per 
unit volume, is 3giv(r/a) l cr;/(2i-\-l).f 
Then adding these two potentials together, we see that the surface inequalities cr; 
produce the same state of strain as would be caused by the bodily forces due to a 
potential — 2(7— l)$w(r/a)'A;/(2t+1), and the surface of the sphere is now subjected to 
no forces. 
This expression is a solid harmonic of the 7 th degree, and therefore the analytical 
* “ On the Bodily Tides of Viscous and Semi-elastic Spheroids, &c.,” Phil. Trans., Part I., 1879, p. 1 
(see § 2). This paper treats of the state of flow of a viscous sphere, hut the problem is exactly the same 
as that concerning elasticity here considered. It is easy to see that if a viscous sphere be deformed into 
the shape of a zonal harmonic, the flow of the fluid must be meridional, and from this we may conclude 
that in the elastic sphere the plane of greatest and least principal stresses is also meridional. This has 
been already assumed to be the case in the present paper. 
t If we could suppose a sphere to have homogeneous elasticity but heterogeneous density, this manner 
of building up the effective disturbing potential would have to be somewhat modified. Such an hypo¬ 
thesis is somewhat absurd, and I shall regard the sphere as homogeneous. In application to the case of 
the earth I shall however pay attention to the smaller density of the superficial layers by halving the 
height of the actual continents and the depth of the actual seas. 
