204 
MR. a. H. DARWIN ON THE STRESSES 
If then the ellipticity e be y(Tooth, the surface and central stress-differences will be 
nearly 1 ton and nearly 8 Ions to the square inch respectively. 
From the Table VII. in § 9 it will appear that cast brass ruptures with a stress - 
difference of about 8 tons to the square inch. 
Thus a spheroid, made of material as strong as brass, and of the same dimensions 
and density as the earth, would only just support an excess or deficiency of ellipticity 
equal to ^jth, above or below the equilibrium ellipticity adapted for its rotation. 
The follow in g is a second example:—If the homogeneous earth (with ellipticity ^ 
were to stop rotating, the stress-difference at the centre would be 33 tons per square 
inch. 
Now suppose the cause of internal stress to be the moon’s tide-generating influence, 
and let m= moon’s mass, and c= moon’s distance. 
Then the potential under which the earth is stressed is —f(m/c 3 )(^— cos 2 0)wr 3 , or 
according to the notation of § 4 — ^(m/c^)wr 2 s :l . 
If we took into account the elastic yielding of the earth and the weight and 
attraction of the tidal protuberance, this potential would have to be diminished. To 
estimate the diminution we must of course know the amount of elastic yielding, but 
as there is no means of approximating thereto, it will be left out of account. 
Then it is obvious that the factor by wdiicli A, as given in (35), must be multiplied 
in order to give the stress-difference is \mwlc\ Thus the surface stress-difference is 
iVi( m / c3 ) wa3 ‘ m absolute force units. 
Then putting M for the earth’s mass, and dividing by gravity, we have 
'i ”y { mCI?J I)iva as the surface value of A in gravitation units. The central value of 
A is of course eight times as great. 
With the numerical data used above, wci= 3605 metric tonnes per square c.m., and 
a/c=jfQ, whence the surface stress-difference is 32 grammes, and the central 
stress-difference 257 grammes per square centimeter. 
But this conclusion is erroneous for the following reason. If we suppose the moon 
to revolve in the terrestrial equator, and imagine that the meridian from which longi¬ 
tudes are measured is the meridian, in which the moon stands at the instant under 
consideration, then the tide-generating potential is — f(m/c 3 )r 2 [^— sin 2 9 cos 2 <£] ; this 
expression may be written f(m/c 3 )r 2 (J— cos 2 d)+|(w/c 3 )r 3 sin 2 dcos 2<j>. The former 
of these terms produces a permanent increase of the earth’s ellipticity, and is confused 
and lost in the ellipticity due to terrestrial rotation, and can produce no stress in the 
earth. The second term is the true tide-generating potential, but it is a sectorial 
harmonic, and I have failed to treat such cases. Now the first of these terms causes 
ellipticity in a homogeneous earth equal to (f«/#)(i m / c3 ) according to the equilibrium 
tide-theory. This ellipticity is equal to T039 X 10“ 6 , an excessively small quantity. 
If however this permanent ellipticity does not exist (and the above investigation in 
reality presumes it not to exist), then there will be a superficial stress-difference equal 
