222 
MR. a. H. DARWIN ON THE STRESSES 
If the order of the harmonic be high, say 30 or 40, we have a regular series of 
mountain chains and intervening valleys running round the sphere in parallels of 
latitude. 
For the sake of convenience I shall always speak as though the equator were a 
region of elevation, but the only effect of changing elevations into depressions, and 
vice versa, is to diametrically reverse the directions of all the stresses. 
The harmonics of the orders 2, 6, 10, &c., have depressions at the poles of the 
sphere; those of orders 4, 8, 12, &c,, have elevations at the pole. 
The harmonic of the fourth order consists of an equatorial continent and a pair of 
circular polar continents, with an intervening depression. That of the sixth order 
consists of an equatorial continent and a pair of annular continents in latitudes 
(about) 60° on one and the other side of the equator. The 8th harmonic brings 
down these new annular continents to about latitude 45°, and adds a pair of polar 
continents ; and so on. 
By a continuation of this process the transition to the mountain chains and valleys 
is obvious. 
In § 5 the case of the 2nd harmonic is considered. As above explained the 
sphere is deformed into a spheroid of revolution. The investigation also applies to 
the case of a rotating spheroid, such as the earth, with either more or less oblateness 
than is appropriate for the figure of equilibrium. 
The lines throughout a meridional section of the spheroid along which the stress- 
difference is constant are shown in Plate 19, fig. 1, and the numbers written on the 
curves give the relative magnitude of the stress-difference. 
It is remarkable that the stress-difference is the same all over the surface. In the 
polar regions the stress-difference diminishes as we descend into the spheroid and then 
increases again; in the equatorial regions it always increases as we descend. The 
maximum value is at the centre, and there the stress-difference is eight times as great 
as at the surface. 
If the elastic solid be highly compressible the stress-differences are not nearly so 
great as on the hypothesis of incompressibility. In all the other cases considered in 
this paper compressibility makes practically no difference in the results. 
On evaluating the stress-difference, on the hypothesis of incompressibility, arising 
from given ellipticity in a spheroid of the size and density of the earth, it appears 
that if the excess or defect of ellipticity above or below the equilibrium value 
(namely -yjy f° r the homogeneous earth) were then the stress-difference at the 
centre would be 8 tons per square inch, and accordingly, if the sphere were made of 
material as strong as brass (see Table VII.), it would be just on the point of rupture. 
Again if the homogeneous earth, with ellipticity yyy, were to stop rotating, the 
central stress-difference would be 33 tons per square inch, and it would rupture if 
made of any material excepting the finest steel. 
