In my paper on “The Gravitational Stability of the Earth,” dynamical arguments were adduced in favour of the hypothesis that the distribution of density within the earth is such that the surfaces of equal density present, in addition to the inequalities depending upon the diurnal rotation, other inequalities which can be specified by spherical harmonics of the first, second, and third degrees. If this is the case, the surface of the earth, by which I mean the surface of the lithosphere, should present corresponding inequalities, and so also should the equipotential surfaces. Analytically, if the density
ρ
is given by an equation of the form
ρ
=
f
0
(
r
) + ϵ
1
f
1
(
r
)S
1
+ ϵ
2
f
2
(
r
)S
2
+ ϵ
3
f
3
(
r
)S
3
, (1) where
f
0
(
r
),
f
1
(
r
), . . . are functions of the distance
r
from the centre, S
1
, S
2
, S
3
are spherical surface harmonics of degrees indicated by the suffixes, and ϵ
1
, ϵ
2
, ϵ
3
are small coefficients, then the surface should have an equation of the form
r
=
a
+
α
1
S
1
+
α
2
S
2
+
α
3
S
3
, (2) where
a
and
α
1
,
α
2
,
α
3
are constants, and the
α
's are small. The elevations and depressions of the lithosphere should be, at least in their main features, expressible by a formula of this type. The actual elevations and depressions are difficult to determine, because all that can be found by observation is the amount of elevation above, or depression below, a particular equipotential surface, the
geoid
, or the surface of the ocean, continued beneath the continents. For a first approximation the potential due to such a distribution of density as is expressed by (1) within a surface expressed by (2) would be given by formulæ of the type V = F
0
(
r
) +
β
1
F
1
(
r
)S
1
+
β
2
F
2
(
r
)S
2
+
β
3
F
3
(
r
)S
3
, (
r
<
a
)
Equipotential surface identification of punctual electrically system charges using measurement results
