218 
ME, G-. H. DARWIN ON THE STRESSES 
It will now be shown that whatever may be the compressibility of the solid, we get 
identically the same solution in the case when the harmonic is of an infinitely high 
order. This is the problem of the harmonic mountains and valleys corrugating a mean 
level surface, which was considered in § 6. The same notation will be adopted here. 
Both i and a are infinite, x becomes ci—(, and —l) — r 2 = a' 2 — r 2 = 2a£ If the 
substitutions here suggested be made in (47) and (48), it will be found that the terms 
with ft) as a factor are multiplied by cii (two infinites), whilst none of the terms with k 
as a factor involve more than one infinite. Hence the latter terms are negligeable 
compared with the former. 
Again i being infinite, K=2i 2 co. Thus if i and a be infinite (47) and (48) reduce to 
cl} W 
df' 
B 
(i2a£) 
cP W 
dP ’ 
mv 
dgdz 
but a/i=b, therefore 
P=6f 
cPW 
d? 5 
R=6f- 
cP W 
clP ’ 
cP W 
d£dz 
•# 
In (36) we found the effective potential, which produced the same state of stress as 
the weight of the mountains whose equation was —£j—h cos z/b ; the result was 
Hence in the present case 
W= —giuh e ^' b cos j 
P ——givk~e 1:0 cos y ~R,-=givlv^<r m cos ^ T =gwh^€~^ ,b 
Sill 
These results are identical with equations (38), which gave the stresses when the 
elastic solid was incompressible. 
It may be concluded from this that, except for the case of the 2nd harmonic 
inequality, compressibility makes very little difference, and for the higher harmonics it 
makes no difference at all. 
II. 
SUMMARY AND DISCUSSION. 
The existence of dry land proves that the earth’s surface is not a figure of 
equilibrium appropriate for the diurnal rotation. 
Hence the interior of the earth must be in a state of stress, and as the land does 
not sink in, nor the sea-bed rise up, the materials of which the earth is made must be 
strong enough to bear this stress. 
We are thus led to inquire how the stresses are distributed in the earth’s mass, and 
what are their magnitudes. These points cannot be discussed without an hypothesis 
as to the interior constitution of the earth. 
