228 MR. G. H. DARWIN ON THE STRESSES 
the continents and sea-beds have sections which are harmonic curves, then if we 
take,— 
The mean level bisecting elevations and depressions as 2480 meters (8150 feet) 
below the sea-level, and the greatest elevation and depression from that mean level as 
3009 meters (9840 feet), it results that the average height of the land above sea-level 
is 350 meters and the average depression of dried sea-bed is 3150 meters. 
It thus appears that 3000 meters would be a proper greatest elevation and de¬ 
pression to assume for the harmonic analysis of this paper, if the earth were 
homogeneous. But as the density of superficial rocks is only a half of the mean 
density of the earth, I shall take 1500 meters as the greatest elevation and depression 
from the mean equilibrium spheroid of revolution. 
It is proper here to note that the height of the undulations of elevation and depres¬ 
sion in the zonal harmonic inequalities is considerably greater towards the poles than 
it is about the equator; it might therefore be maintained that by making 1500 meters 
the equatorial height, we are taking too high an estimate. But the state of stress 
caused in the sphere at any point depends very much more on the height of the 
inequality in the neighbourhood of a superficial point immediately over the point 
considered, than it does on the inequalities in remote parts of the sphere. 
Now in all the inequalities, except the 2nd harmonic, I have considered the state of 
stress in the equatorial region, and it will therefore I think be proper to adhere to the 
1500 meters for the greatest height and depression. 
We have next to consider, what order of harmonic inequalities is most nearly 
analogous to the great terrestrial continents and oceans. The most obvious case to 
take is that of the two Americas and Africa with Europe. The average longitude of 
the Americas is between 60° and 80° W., and the average longitude of Africa is about 
25° E., hence there is a difference of longitude of about a right-angle between the two 
masses. These two great continents would be more nearly represented by an harmonic 
of the sectorial class,* rather than by a zonal harmonic, nevertheless I think the 
solution for the zonal harmonic will be adequate for the present purpose. 
Now it has been explained above that the harmonic of the fourth order represents 
an equatorial continent and a pair of polar continents. In the case of the 4th 
harmonic therefore there is a right angle of a great circle between contiguous con¬ 
tinents. We may conclude from this that the large terrestrial inequalities are about 
equivalent to the harmonic of the fourth order. 
Table V. (b), § 7, gives the maximum stress-differences under the centre of the equa¬ 
torial elevation of the several zonal harmonics, the height of each being 1500 meters. 
* The sectorial harmonic of the fourth order sin 4 0 cos 40 would well represent these two great con¬ 
tinents. It would fairly represent China and Australia; but would annihilate the Himalayan plateau, 
and place another great continent in mid-Pacific. It is not at all difficult to find the stress-difference 
under the centre of a sectorial inequality, but to find it generally involves the solution of a cubic 
equation. 
