where
/1 + [epsilon] squared 1 \
A = 2[pi]k squared[rho]( ------------- tan^(-1) [epsilon] - -------- ),
\ [epsilon] cubed [epsilon] squared/
/1 + [epsilon] squared 1 + [epsilon] squared \
C = 4[pi]k squared[rho]( -------------- - ------------- tan^(-1) [epsilon] ),
\ [epsilon] squared [epsilon] cubed /
and k squared the constant of attraction. Besides the attraction of the mass
of the ellipsoid, the centrifugal force at P has for components +
x[omega] squared, + y[omega] squared, 0; then the condition of fluid equilibrium is
(A - [omega] squared)xdx + (A - [omega] squared)ydy + Czdz = 0,
which by integration gives
(A - [omega] squared)(x squared + y squared) + Cz squared = constant.
This is the equation of an ellipsoid of rotation, and therefore the
equilibrium is possible. The equation coincides with that of the
surface of the fluid mass if we make
A - [omega] squared = C/(1 + [epsilon] squared),
which gives
[omega] squared 3 + [epsilon] squared 3
------------ = -------------- tan^(-1) [epsilon] - ---------- .
2[pi]k squared[rho] [epsilon] cubed [epsilon] squared
In the case of the earth, which is nearly spherical, we obtain by
expanding the expression for [omega] squared in powers of [epsilon] squared,
rejecting the higher powers, and remarking that the ellipticity e =
1/2[epsilon] squared,
[omega] squared/2[pi]k squared[rho] = 4[epsilon] squared/15 = 8e/15.
Now if m be the ratio of the centrifugal force to the intensity of
gravity at the equator, and a = c(1 + e), then
m = a[omega] squared,/(4/3)[pi]k squared[rho]a, :. [omega] squared/2[pi]k squared[rho] = (2/3)m.
In the case of the earth it is a matter of observation that m =
1/289, hence the ellipticity
e = 5m/4 = 1/231,
so that the ratio of the axes on the supposition of a homogeneous
fluid earth is 230:231, as stated by Newton.
Now, to come to the case of a heterogeneous fluid, we shall assume
that its surfaces of equal density are spheroids, concentric and
having a common axis of rotation, and that the ellipticity of these
surfaces varies from the centre to the outer surface, the density also
varying. In other words, the body is composed of homogeneous
spheroidal shells of variable density and ellipticity. On this
supposition we shall express the attraction of the mass upon a
particle in its interior, and then, taking into account the
centrifugal force, form the equation expressing the condition of fluid
equilibrium. The attraction of the homogeneous spheroid x squared + y squared + z squared(1
+ 2e) = c squared(1 + 2e), where e is the ellipticity (of which the square is
neglected), on an internal particle, whose co-ordinates are x = f, y =
0, z = h, has for its x and z components
X' = -(4/3)[pi]k squared[rho]f(1 - (2/5)e),
Z' = -(4/3)[pi]k squared[rho]h(1 + (4/5)e),
the Y component being of course zero. Hence we infer that the
attraction of a shell whose inner surface has an ellipticity e, and
its outer surface an ellipticity e + de, the density being [rho], is
expressed by
dX' = (4/3).(2/5)[pi]k squared[rho]f de, dZ' = -(4/3).(4/5)[pi]k squared[rho]h de.
To apply this to our heterogeneous spheroid; if we put c1 for the
semiaxis of that surface of equal density on which is situated the
attracted point P, and c0 for the semiaxis of the outer surface, the
attraction of that portion of the body which is exterior to P, namely,
of all the shells which enclose P, has for components
_ _
8 /c0 de 16 /c0 de
X0 = --[pi]k squaredf | [rho] --dc, Z0 = - -- [pi]k squaredh | [rho] --dc,
15 _/c1 dc 15 _/c1 dc
both e and [rho] being functions of c. Again the attraction of a
homogeneous spheroid of density [rho] on an _external_ point f, h has
the components
X" = -(4/3)[pi]k squared[rho]fr^(-3) {c cubed(1 + 2e) - [lambda]ec^5},
Z" = -(4/3)[pi]k squared[rho]hr^(-3) {c cubed(1 + 2e) - [lambda]'ec^5},
where [lambda] = (3/5)(4h squared - f squared)/r^4,
[lambda]' = (3/5)(2h squared - 3f squared)/r^4, and r squared = f squared + h squared.
Now e being considered a function of c, we can at once express the
attraction of a shell (density [rho]) contained between the surface
defined by c + dc, e + de and that defined by c, e upon an external
point; the differentials with respect to c, viz. dX" dZ", must then
be integrated with [rho] under the integral sign as being a function
of c. The integration will extend from c = 0 to c = c1. Thus the
components of the attraction of the heterogeneous spheroid upon a
particle within its mass, whose co-ordinates are f, 0, h, are
_ _
4 | 1 /c1
X = - -- [pi]k squaredf | -- | [rho] d{c cubed(1 + 2e)}
3 |_r cubed _/0
_ _ _
[lambda] /c1 2 /c1 |
- -------- | [rho] d(ec^5) - -- | [rho] de |,
r cubed _/0 5 _/0 _|
_ _
4 | 1 /c1
Z = - -- [pi]k squaredh | -- | [rho] d{c cubed(1 + 2e)}
3 |_r cubed _/0
_ _ _
[lambda]'/c1 4 /c1 |
- -------- | [rho] d(ec^5) + -- | [rho] de |.
r cubed _/0 5 _/0 _|
We take into account the rotation of the earth by adding the
centrifugal force f[omega] squared = F to X. Now, the surface of constant
density upon which the point f, 0, h is situated gives (1 - 2e) fdf +
hdh = 0; and the condition of equilibrium is that (X + F)df + Zdh = 0.
Therefore,
(X + F)h = Zf(1 - 2e),
which, neglecting small quantities of the order e squared and putting
[omega] squaredt squared = 4[pi] squaredk squared, gives
_ _ _
2e /c1 6 /c1 6 /c0 3[pi]
-- | [rho]d{c cubed(1 + 2e)} - ---- | [rho]d(ec^5) - -- | [rho]de = -----.
r cubed_/0 5r^5 _/0 5 _/c1 t squared
Here we must now put c for c1, c for r; and 1 + 2e under the first
integral sign may be replaced by unity, since small quantities of the
second order are neglected. Two differentiations lead us to the
following very important differential equation (Clairault):
d squarede 2[rho]c squared de / 2[rho]c 6 \
--- + -------------- . -- + ( -------------- - -- ) e = 0.
dc squared [int][rho]c squareddc dc \[int][rho]c squareddc c squared/
When [rho] is expressed in terms of c, this equation can be
integrated. We infer then that a rotating spheroid of very small
ellipticity, composed of fluid homogeneous strata such as we have
specified, will be in equilibrium; and when the law of the density is
expressed, the law of the corresponding ellipticities will follow.
If we put M for the mass of the spheroid, then
_
4[pi] /c c cubed 4[pi] squared
M = ----- | [rho]d{c cubed(1 + 2e)}; and m = --- . -----,
3 _/0 M t squared
and putting c = c0 in the equation expressing the condition of
equilibrium, we find
_
4 6 /c
M(2e - m) = -- [pi].--- | [rho]d(ec^5).
3 5c squared_/0
Making these substitutions in the expressions for the forces at the
surface, and putting r/c = 1 + e - e(h/c) squared, we get
_ _
Mk squared | 3 /5 \ h squared | f
G cos [phi] = --- | 1 - e - -- m + ( - m - 2e) -- | --
ac |_ 2 \2 / c squared_| c
_ _
Mk squared | 3 /5 \ h squared | h
G sin [phi] = --- | 1 + e - -- m + ( - m - 2e) -- | --.
ac |_ 2 \2 / c squared_| c
Here G is gravity in the latitude [phi], and a the radius of the
equator. Since
sec [phi] = (c/f){1 + e + (eh squared/c squared)},
_ _
Mk squared | 3 /5 \ |
G = --- | 1 - -- m + ( -- m - e) sin squared [phi] |,
ac |_ 2 \2 / _|
an expression which contains the theorems we have referred to as
discovered by Clairault.
The theory of the figure of the earth as a rotating ellipsoid has been
especially investigated by Laplace in his _Mecanique celeste_. The
principal English works are:--Sir George Airy, _Mathematical Tracts_,
a lucid treatment without the use of Laplace's coefficients;
Archdeacon Pratt's _Attractions and Figure of the Earth_; and
O'Brien's _Mathematical Tracts_; in the last two Laplace's
coefficients are used.
In 1845 Sir G.G. Stokes (_Camb. Trans._ viii.; see also _Camb. Dub.
Math. Journ._, 1849, iv.) proved that if the external form of the
sea--imagined to percolate the land by canals--be a spheroid with small
ellipticity, then the law of gravity is that which we have shown above;
his proof required no assumption as to the ellipticity of the internal
strata, or as to the past or present fluidity of the earth. This
investigation admits of being regarded conversely, viz. as determining
the elliptical form of the earth from measurements of gravity; if G, the
observed value of gravity in latitude [phi], be expressed in the form G
= g(1 + ss sin squared [phi]), where g is the value at the equator and ss a
coefficient. In this investigation, the square and higher powers of the
ellipticity are neglected; the solution was completed by F.R. Helmert
with regard to the square of the ellipticity, who showed that a term
with sin squared 2[phi] appeared (see Helmert, _Geodaesie_, ii. 83). For the
coefficient of this term, the gravity measurements give a small but not
sufficiently certain value; we therefore assume a value which agrees
best with the hypothesis of the fluid state of the entire earth; this
assumption is well supported, since even at a depth of only 50 km. the
pressure of the superincumbent crust is so great that rocks become
plastic, and behave approximately as fluids, and consequently the crust
of the earth floats, to some extent, on the interior (even though this
may not be fluid in the usual sense of the word). This is the geological
theory of "Isostasis" (cf. GEOLOGY); it agrees with the results of
measurements of gravity (_vide infra_), and was brought forward in the
middle of the 19th century by J.H. Pratt, who deduced it from
observations made in India.
The sin squared 2[phi] term in the expression for G, and the corresponding
deviation of the meridian from an ellipse, have been analytically
established by Sir G.H. Darwin and E. Wiechert; earlier and less
complete investigations were made by Sir G.B. Airy and O. Callandreau.
In consequence of the sin squared 2[phi] term, two parameters of the level
surfaces in the interior of the earth are to be determined; for this
purpose, Darwin develops two differential equations in the place of the
one by Clairault. By assuming Roche's law for the variation of the
density in the interior of the Earth, viz. [rho] = [rho]1 - k(c/c1) squared, k
being a coefficient, it is shown that in latitude 45 deg., the meridian is
depressed about 31/4 metres from the ellipse, and the coefficient of the
term sin squared [phi] cos squared [phi] (= 1/4 sin squared 2[phi]) is -0.0000295. According to
Wiechert the earth is composed of a kernel and a shell, the kernel being
composed of material, chiefly metallic iron, of density near 8.2, and
the shell, about 900 miles thick, of silicates, &c., of density about
3.2. On this assumption the depression in latitude 45 deg. is 23/4 metres, and
the coefficient of sin squared [phi] cos squared [phi] is, in round numbers,
-0.0000280.[2] To this additional term in the formula for G, there
corresponds an extension of Clairault's formula for the calculation of
the flattening from ss with terms of the higher orders; this was first
accomplished by Helmert.
For a long time the assumption of an ellipsoid with three unequal axes
has been held possible for the figure of the earth, in consequence of an
important theorem due to K.G. Jacobi, who proved that for a homogeneous
fluid in rotation a spheroid is not the only form of equilibrium; an
ellipsoid rotating round its least axis may with certain proportions of
the axes and a certain time of revolution be a form of equilibrium.[3]
It has been objected to the figure of three unequal axes that it does
not satisfy, in the proportions of the axes, the conditions brought out
in Jacobi's theorem (c: a < 1/[root]2). Admitting this, it has to be
noted, on the other hand, that Jacobi's theorem contemplates a
homogeneous fluid, and this is certainly far from the actual condition
of our globe; indeed the irregular distribution of continents and oceans
suggests the possibility of a sensible divergence from a perfect surface
of revolution. We may, however, assume the ellipsoid with three unequal
axes to be an interpolation form. More plausible forms are little
adapted for computation.[4] Consequently we now generally take the
ellipsoid of rotation as a basis, especially so because measurements of
gravity have shown that the deviation from it is but trifling.
_Local Attraction._
In speaking of the figure of the earth, we mean the surface of the sea
imagined to percolate the continents by canals. That this surface
should turn out, after precise measurements, to be exactly an ellipsoid
of revolution is _a priori_ improbable. Although it may be highly
probable that originally the earth was a fluid mass, yet in the cooling
whereby the present crust has resulted, the actual solid surface has
been left most irregular in form. It is clear that these irregularities
of the visible surface must be accompanied by irregularities in the
mathematical figure of the earth, and when we consider the general
surface of our globe, its irregular distribution of mountain masses,
continents, with oceans and islands, we are prepared to admit that the
earth may not be precisely any surface of revolution. Nevertheless,
there must exist some spheroid which agrees very closely with the
mathematical figure of the earth, and has the same axis of rotation. We
must conceive this figure as exhibiting slight departures from the
spheroid, the two surfaces cutting one another in various lines; thus a
point of the surface is defined by its latitude, longitude, and its
height above the "spheroid of reference." Calling this height N, then of
the actual magnitude of this quantity we can generally have no
information, it only obtrudes itself on our notice by its variations. In
the vicinity of mountains it may change sign in the space of a few
miles; N being regarded as a function of the latitude and longitude, if
its differential coefficient with respect to the former be zero at a
certain point, the normals to the two surfaces then will lie in the
prime vertical; if the differential coefficient of N with respect to the
longitude be zero, the two normals will lie in the meridian; if both
coefficients are zero, the normals will coincide. The comparisons of
terrestrial measurements with the corresponding astronomical
observations have always been accompanied with discrepancies. Suppose A
and B to be two trigonometrical stations, and that at A there is a
disturbing force drawing the vertical through an angle [delta], then it
is evident that the apparent zenith of A will be really that of some
other place A', whose distance from A is r[delta], when r is the earth's
radius; and similarly if there be a disturbance at B of the amount
[delta]', the apparent zenith of B will be really that of some other
place B', whose distance from B is r[delta]'. Hence we have the
discrepancy that, while the geodetic measurements deal with the points A
and B, the astronomical observations belong to the points A', B'. Should
[delta], [delta]' be equal and parallel, the displacements AA', BB' will
be equal and parallel, and no discrepancy will appear. The
non-recognition of this circumstance often led to much perplexity in the
early history of geodesy. Suppose that, through the unknown variations
of N, the probable error of an observed latitude (that is, the angle
between the normal to the mathematical surface of the earth at the given
point and that of the corresponding point on the spheroid of reference)
be [epsilon], then if we compare two arcs of a degree each in mean
latitudes, and near each other, say about five degrees of latitude
apart, the probable error of the resulting value of the ellipticity will
be approximately +- 1/500[epsilon], [epsilon] being expressed in
seconds, so that if [epsilon] be so great as 2" the probable error of
the resulting ellipticity will be greater than the ellipticity itself.
It is necessary at times to calculate the attraction of a mountain, and
the consequent disturbance of the astronomical zenith, at any point
within its influence. The deflection of the plumb-line, caused by a
local attraction whose amount is k squaredA[delta], is measured by the ratio of
k squaredA[delta] to the force of gravity at the station. Expressed in seconds,
the deflection [Lambda] is
[Lambda] = 12".447A[delta]/[rho],
where [rho] is the mean density of the earth, [delta] that of the
attracting mass, and A = [f]s^(-3)xdv, in which dv is a volume element
of the attracting mass within the distance s from the point of
deflection, and x the projection of s on the horizontal plane through
this point, the linear unit in expressing A being a mile. Suppose, for
instance, a table-land whose form is a rectangle of 12 miles by 8 miles,
having a height of 500 ft. and density half that of the earth; let the
observer be 2 miles distant from the middle point of the longer side.
The deflection then is 1".472; but at 1 mile it increases to 2".20.
At sixteen astronomical stations in the English survey the disturbance
of latitude due to the form of the ground has been computed, and the
following will give an idea of the results. At six stations the
deflection is under 2", at six others it is between 2" and 4", and at
four stations it exceeds 4". There is one very exceptional station on
the north coast of Banffshire, near the village of Portsoy, at which the
deflection amounts to 10", so that if that village were placed on a map
in a position to correspond with its astronomical latitude, it would be
1000 ft. out of position! There is the sea to the north and an
undulating country to the south, which, however, to a spectator at the
station does not suggest any great disturbance of gravity. A somewhat
rough estimate of the local attraction from external causes gives a
maximum limit of 5", therefore we have 5" which must arise from
unequal density in the underlying strata in the surrounding country. In
order to throw light on this remarkable phenomenon, the latitudes of a
number of stations between Nairn on the west, Fraserburgh on the east,
and the Grampians on the south, were observed, and the local deflections
determined. It is somewhat singular that the deflections diminish in all
directions, not _very_ regularly certainly, and most slowly in a
south-west direction, finally disappearing, and leaving the maximum at
the original station at Portsoy.
The method employed by Dr C. Hutton for computing the attraction of
masses of ground is so simple and effectual that it can hardly be
improved on. Let a horizontal plane pass through the given station; let
r, [theta] be the polar co-ordinates of any point in this plane, and r,
[theta], z, the co-ordinates of a particle of the attracting mass; and
let it be required to find the attraction of a portion of the mass
contained between the horizontal planes z = 0, z = h, the cylindrical
surfaces r = r1, r = r2, and the vertical planes [theta] = [theta]1,
[theta] = [theta]2. The component of the attraction at the station or
origin along the line [theta] = 0 is
_ _ _
/r2 /[theta]2 /h r squaredcos [theta]
k squared[delta] | | | ------------- dr d[theta] dz
_/r1 _/[theta]1 _/0 (r squared+z squared)^(3/2)
= k squared[delta]h(sin[theta]2 - sin[theta]1) log{r2 + (r2 squared + h squared)^(1/2)/r1 + (r1 squared + h squared)^(1/2)}.
By taking r2 - r1, sufficiently small, and supposing h also small
compared with r1 + r2 (as it usually is), the attraction is
k squared[delta](r2 - r1)(sin [theta]2 - sin [theta]1)h/r,
where r= 1/2(r1 + r2). This form suggests the following procedure. Draw on
the contoured map a series of equidistant circles, concentric with the
station, intersected by radial lines so disposed that the sines of their
azimuths are in arithmetical progression. Then, having estimated from
the map the mean heights of the various compartments, the calculation is
obvious.
In mountainous countries, as near the Alps and in the Caucasus,
deflections have been observed to the amount of as much as 30", while
in the Himalayas deflections amounting to 60" were observed. On the
other hand, deflections have been observed in flat countries, such as
that noted by Professor K.G. Schweizer, who has shown that, at certain
stations in the vicinity of Moscow, within a distance of 16 miles the
plumb-line varies 16" in such a manner as to indicate a vast deficiency
of matter in the underlying strata; deflections of 10" were observed in
the level regions of north Germany.
Since the attraction of a mountain mass is expressed as a numerical
multiple of [delta] : [rho] the ratio of the density of the mountain to
that of the earth, if we have any independent means of ascertaining the
amount of the deflection, we have at once the ratio [rho]:[delta], and
thus we obtain the mean density of the earth, as, for instance, at
Schiehallion, and afterwards at Arthur's Seat. Experiments of this kind
for determining the mean density of the earth have been made in greater
numbers; but they are not free from objection (see GRAVITATION).
Let us now consider the perturbation attending a spherical subterranean
mass. A compact mass of great density at a small distance under the
surface of the earth will produce an elevation of the mathematical
surface which is expressed by the formula
y = a mu((1 - 2u cos [theta] + u squared)^(-1/2) - 1),
where a is the radius of the (spherical) earth, a(1 - u) the distance
of the disturbing mass below the surface, mu the ratio of the disturbing
mass to the mass of the earth, and a[theta] the distance of any point on
the surface from that point, say Q, which is vertically over the
disturbing mass. The maximum value of y is at Q, where it is y = a muu(1
-u). The deflection at the distance a[theta] is [Lambda] = muu
sin[theta](1 - 2u cos[theta] + u squared)^(-3/2), or since [theta] is small,
putting h + u = 1, we have [Lambda] = mu[theta](h squared + [theta] squared)^(-3/2).
The maximum deflection takes place at a point whose distance from Q is
to the depth of the mass as 1:[root]2, and its amount is 2 mu/3
[root](3h squared). If, for instance, the disturbing mass were a sphere a mile
in diameter, the excess of its density above that of the surrounding
country being equal to half the density of the earth, and the depth of
its centre half a mile, the greatest deflection would be 5", and the
greatest value of y only two inches. Thus a large disturbance of gravity
may arise from an irregularity in the mathematical surface whose actual
magnitude, as regards height at least, is extremely small.
The effect of the disturbing mass mu on the vibrations of a pendulum
would be a maximum at Q; if v be the number of seconds of time gained
per diem by the pendulum at Q, and [sigma] the number of seconds of
angle in the maximum deflection, then it may be shown that v/[sigma] =
[pi][root]3/10.
The great Indian survey, and the attendant measurements of the degree of
latitude, gave occasion to elaborate investigations of the deflection of
the plumb-line in the neighbourhood of the high plateaus and mountain
chains of Central Asia. Archdeacon Pratt (_Phil. Trans._, 1855 and
1857), in instituting these investigations, took into consideration the
influence of the apparent diminution of the mass of the earth's crust
occasioned by the neighbouring ocean-basins; he concluded that the
accumulated masses of mountain chains, &c., corresponded to subterranean
mass diminutions, so that over any level surface in a fixed depth
(perhaps 100 miles or more) the masses of prisms of equal section are
equal. This is supported by the gravity measurements at More in the
Himalayas at a height of 4696 metres, which showed no deflection due to
the mountain chain (_Phil. Trans._, 1871); more recently, H.A. Faye
(_Compt. rend._, 1880) arrived at the same conclusion for the entire
continent.
This compensation, however, must only be regarded as a general
principle; in certain cases, the compensating masses show marked
horizontal displacements. Further investigations, especially of gravity
measurements, will undoubtedly establish other important facts. Colonel
S.G. Burrard has recently recalculated, with the aid of more exact data,
certain Indian deviations of the plumb-line, and has established that in
the region south of the Himalayas (lat. 24 deg.) there is a subterranean
perturbing mass. The extent of the compensation of the high mountain
chains is difficult to recognize from the latitude observations, since
the same effect may result from different causes; on the other hand,
observations of geographical longitude have established a strong
compensation.[5]
_Meridian Arcs._
The astronomical stations for the measurement of the degree of latitude
will generally lie not exactly on the same meridian; and it is therefore
necessary to calculate the arcs of meridian M which lie between the
latitude of neighbouring stations. If S be the geodetic line calculated
from the triangulation with the astronomically determined azimuths
[alpha]1 and [alpha]2, then
_ _
cos [alpha] | 1 S squared |
M = S ------------------- | 1 + -- -------- sin squared [alpha] ... |,
cos 1/2[Delta][alpha] |_ 12 [alpha] squared _|
in which 2[alpha] = [alpha]1 + [alpha]2 - 180 deg., [Delta][alpha] =
[alpha]2 - [alpha]1 - 180 deg..
The length of the arc of meridian between the latitudes [phi]1 and
[phi]2 is
_[phi]2 _[phi]2
/ / (1 - e squared)d[phi]
M = | [rho]d[phi] = [alpha] | -----------------------
_/ _/ (1 - e squaredsin squared[phi])^(3/2)
[phi]1 [phi]1
where a squarede squared = a squared - b squared; instead of using the eccentricity e, put the ratio
of the axes b:a = 1 - n:1 + n, then
_[phi]2
/ b(1 + n)(1 - n squared)d[phi]
M = | ------------------------------.
_/ (1 + 2n cos 2[phi] + n squared)^(3/2)
[phi]1