PRESIDENTIAL ADDRESS. 431 
The force that keeps the Pacific Ocean on one side of the earth is gravity, directed 
more towards the centre of gravity than the centre of figure. An adequate cause 
for the eccentric position of the centre of gravity is found in the necessary state of 
aggregation which the earth must have had if at one time it was as compressible 
as granite. The theory of gravitational instability accounts for the existence of 
the Pacitic Ocean. 
But we can go much further than this in the direction of accounting for t 
continental and oceanic regions. We keep in mind the eccentric position of the 
centre of gravity, and try to discover the effect of rotation upon a planet of which 
the centre of gravity does uot coincide with the centre of figure. The shape of a 
rotating planet must be nearly an oblate spheroid; but the figure of the ocean 
would, owing to its greater mobility, be rather more protuberant at the equator 
than the figure of the planet on which it rests. The primary effect of the rotation 
of the earth upon the distribution of continent and ocean is to draw the ocean 
towards the equator, so as to tend to expose the arctic and antarctic regions. We 
have seen that both arctic and antarctic are parts of the continental region. But 
there is an important secondary effect. Under the influence of the rotation the 
parts of greater density tend to recede further from the axis than the parts of less 
density. Ifthe density is greater in one hemispheroid than in the other, so that 
the position of the centre of gravity is eccentric, the effect must be to produce a 
sort of furrowed surface; and the amount of elevation and depression so produced 
can be described by an exact mathematical formula. It has been proved that this 
formula is the sort of expression which mathematicians name a spherical harmonic 
of the third degree. 
The shape of the earth is also influenced by another circumstance. We know 
that at one time the moon was much nearer to the earth than it is now, and that 
the two bodies once rotated about their common centre of gravity almost as a 
single rigid system. The month was nearly as short as the day, and the moon 
was nearly fixed in the sky. The earth must then have been drawn out towards 
the moon, so that its surface was more nearly an ellipsoid with three unequal axes 
than it is now. The primary effect of the ellipsoidal condition upon the distribu- 
tion of continent and ocean would be to raise the surface above the ocean near the 
opposite extremities of the greatest diameter of the equator. But, again, owing to 
the eccentric position of the centre of gravity, there would be an important 
secondary effect. The gravitational attraction of an ellipsoid differs from that of 
a sphere, and it may be represented as the attraction of a sphere together with 
an additional attraction. If the density was greater in one hemi-ellipsoid than in 
the other, the additional attraction would produce a greater effect in the parts 
where the density was in excess, and the result, just as in the case of rotation, 
would be a furrowing of the surface. It has been proved that the formula for this 
furrowing also is expressed by a spherical harmonic of the third degree. 
We are brought to the theory of spherical harmonics and the spherical 
harmonic analysis. Spherical harmonics are certain quantities which vary in a 
regular fashion over the surface of a sphere, becoming positive in some parts and 
negative in others, I spoke just now of making a model of a nearly spherical 
surface by removing material from some parts and heaping it up on others. 
Spherical harmonics specify standard patterns of deformation of spheres. For 
instance, we might remove material over one hemisphere down to the surface of an 
equal but not concentric sphere (cf. fig. 5) and heap up the material over the 
other hemisphere. We should produce a sphere equal to the original but in a new 
position. The formula for the thickness of the material removed or added is a 
spherical harmonic of the first degree. It specifies the simplest standard pattern 
of deformation. Again, we might remove material from some parts of our model 
and heap it up on other parts so as to convert the sphere into an ellipsoid. The 
formula for the thickness of that which is removed or added is aspherical harmonic 
of the second degree. Deformation of a sphere into an ellipsoid is the second 
standard pattern of deformation. The mathematical method of determining the 
appropriate series of standard patterns is the theory of spherical harmonics. Its 
importance arises from the result that any pattern whatever can be reached by 
