Chapter IX: Part 9

We may here briefly indicate the line of reasoning by which some of
the most important results may be obtained. If X, Y, Z be the
components parallel to three rectangular axes of the forces acting on
a particle of a fluid mass at the point x, y, z, then, p being the
pressure there, and [rho] the density,

dp = [rho](Xdx + Ydy + Zdz);

and for equilibrium the necessary conditions are, that [rho](Xdx + Ydy
+ Zdz) be a complete differential, and at the free surface Xdx + Ydy +
Zdz = 0. This equation implies that the resultant of the forces is
normal to the surface at every point, and in a homogeneous fluid it is
obviously the differential equation of all surfaces of equal pressure.
If the fluid be heterogeneous then it is to be remarked that for
forces of attraction according to the ordinary law of gravitation, if
X, Y, Z be the components of the attraction of a mass whose potential
is V, then

dV dV dV
Xdx + Ydy + Zdz = --dx + --dy + --dz,
dx dy dz

which is a complete differential. And in the case of a fluid rotating
with uniform velocity, in which the so-called centrifugal force enters
as a force acting on each particle proportional to its distance from
the axis of rotation, the corresponding part of Xdx + Ydy + Zdz is
obviously a complete differential. Therefore for the forces with which
we are now concerned Xdx + Ydy + Zdz = dU, where U is some function of
x, y, z, and it is necessary for equilibrium that dp = [rho]dU be a
complete differential; that is, [rho] must be a function of U or a
function of p, and so also p a function of U. So that dU = 0 is the
differential equation of surfaces of equal pressure and density.

We may now show that a homogeneous fluid mass in the form of an oblate
ellipsoid of revolution having a uniform velocity of rotation can be
in equilibrium. It may be proved that the attraction of the ellipsoid
x squared + y squared + z squared(1 + [epsilon] squared) = c squared(1 + [epsilon] squared); upon a particle P
of its mass at x, y, z has for components

X = -Ax, Y = -Ay, Z = -Cz,

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

This, after integration, gives

/ 5 5 \ / 21 \
M/b = ( 1 + n + -n squared + -n cubed) [alpha]0 - ( 3n + 3n squared + --n cubed) [alpha]1
\ 4 4 / \ 8 /

/15 15 \ /35 \
+ ( --n squared + --n cubed) [alpha]2 - ( --n cubed ) [alpha]3,
\ 8 8 / \24 /

where

[alpha]0 = [phi]2 - [phi]1
[alpha]1 = sin ([phi]2 - [phi]1) cos ([phi]2 + [phi]1)
[alpha]2 = sin 2([phi]2 - [phi]1) cos 2([phi]2 + [phi]1)
[alpha]3 = sin 3([phi]2 - [phi]1) cos 3([phi]2 + [phi]1).

The part of M which depends on n cubed is very small; in fact, if we calculate it for one of the longest arcs measured, the Russian arc, it amounts to only an inch and a half, therefore we omit this term, and put for M/b the value

/ 5 \ /15 \
(l + n + --n squared) [alpha]0 - (3n + 3n squared) [alpha]1 + ( --n squared) [alpha]2.
\ 4 / \ 8 /

Now, if we suppose the observed latitudes to be affected with errors, and that the true latitudes are [phi]1 + x1, [phi]2 + x2; and if further we suppose that n1 + dn is the true value of a - b:a + b, and that n1 itself is merely a very approximate numerical value, we get, on making these substitutions and neglecting the influence of the corrections x on the _position_ of the arc in latitude, i.e. on [phi]1 + [phi]2,

/ 5 \ / \ /15 \
M/b = ( 1 + n + --n1 squared)[alpha]0 - (3n1 + 3n1 squared)[alpha]1 + ( --n1 squared)[alpha]2
\ 4 / \ / \8 /
_ _
| / 5 \ / \ /15 \ |
+ | ( 1 + -- n1 )a0 - ( 3 + 6n1 )a1 + ( -- n1 )a2 | dn
|_ \ 2 / \ / \4 / _|
_ _
| da1 |
+ | 1 + n1 - 3n --- | da0;
|_ da0_|

here da0 = x2 - x1; and as b is only known approximately, put b = b1(1 + u); then we get, after dividing through by the coefficient of da0, which is = 1 + n1 - 3n1 cos([phi]2 - [phi]1) cos([phi]2 + [phi]1), an equation of the form x2 = x1 + h + fu + gv, where for convenience we put v for dn.

Now in every measured arc there are not only the extreme stations determined in latitude, but also a number of intermediate stations so that if there be i + 1 stations there will be i equations

x2 = x1 + f1u + g1v + h1
x3 = x1 + f2u + g2v + h2
: : :
: : :
x_i = x1 + f_iu + g_iv + h_i

In combining a number of different arcs of meridian, with the view of determining the figure of the earth, each arc will supply a number of equations in u and v and the corrections to its observed latitudes. Then, according to the method of least squares, those values of u and v are the most probable which render the sum of the squares of _all_ the errors x a minimum. The corrections x which are here applied arise not from errors of observation only. The mere uncertainty of a latitude, as determined with modern instruments, does not exceed a very small fraction of a second as far as errors of observation go, but no accuracy in observing will remove the error that may arise from local attraction. This, as we have seen, may amount to some seconds, so that the corrections x to the observed latitudes are attributable to local attraction. Archdeacon Pratt objected to this mode of applying least squares first used by Bessel; but Bessel was right, and the objection is groundless. Bessel found, in 1841, from ten meridian arcs with a total amplitude of 50 deg..6:

a = 3272077 toises = 6377397 metres.
e (ellipticity) = (a - b)/a = 1/299.15 (prob. error +- 3.2).

The probable error in the length of the earth's quadrant is +- 336 m.

We now give a series of some meridian-arcs measurements, which were utilized in 1866 by A.R. Clarke in the _Comparisons of the Standards of Length_, pp. 280-287; details of the calculations are given by the same author in his _Geodesy_ (1880), pp. 311 et seq.

The data of the French arc from Formentera to Dunkirk are--

Stations. Astronomical Distance of
Latitudes. Parallels.
deg. ' " Ft.
Formentera 38 39 53.17 ..
Mountjouy 41 21 44.96 982671.04
Barcelona 41 22 47.90 988701.92
Carcassonne 43 12 54.30 1657287.93
Pantheon 48 50 47.98 3710827.13
Dunkirk 51 2 8.41 4509790.84

The distance of the parallels of Dunkirk and Greenwich, deduced from the extension of the triangulation of England into France, in 1862, is 161407.3 ft., which is 3.9 ft. greater than that obtained from Captain Kater's triangulation, and 3.2 ft. less than the distance calculated by Delambre from General Roy's triangulation. The following table shows the data of the English arc with the distances in standard feet from Formentera.

deg. ' " Ft.
Formentera .. ..
Greenwich 51 28 38.30 4671198.3
Arbury 52 13 26.59 4943837.6
Clifton 53 27 29.50 5394063.4
Kellie Law 56 14 53.60 6413221.7
Stirling 57 27 49.12 6857323.3
Saxavord 60 49 37.21 8086820.7

The latitude assigned in this table to Saxavord is not the directly observed latitude, which is 60 deg. 49' 38.58", for there are here a cluster of three points, whose latitudes are astronomically determined; and if we transfer, by means of the geodesic connexion, the latitude of Gerth of Scaw to Saxavord, we get 60 deg. 49' 36.59"; and if we similarly transfer the latitude of Balta, we get 60 deg. 49' 36.46". The mean of these three is that entered in the above table.

For the Indian arc in long. 77 deg. 40' we have the following data:--

deg. ' " Ft.
Punnea 8 9 31.132 ..
Putchapolliam 10 59 42.276 1029174.9
Dodagunta 12 59 52.165 1756562.0
Namthabad 15 5 53.562 2518376.3
Daumergida 18 3 15.292 3591788.4
Takalkhera 21 5 51.532 4697329.5
Kalianpur 24 7 11.262 5794695.7
Kaliana 29 30 48.322 7755835.9

The data of the Russian arc (long. 26 deg. 40') taken from Struve's work are as below:--

deg. ' " Ft.
Staro Nekrasovsk 45 20 2.94 ..
Vodu-Luy 47 1 24.98 616529.81
Suprunkovzy 48 45 3.04 1246762.17
Kremenets 50 5 49.95 1737551.48
Byelin 52 2 42.16 2448745.17
Nemesh 54 39 4.16 3400312.63
Jacobstadt 56 30 4.97 4076412.28
Dorpat 58 22 47.56 4762421.43
Hogland 60 5 9.84 5386135.39
Kilpi-maki 62 38 5.25 6317905.67
Tornea 65 49 44.57 7486789.97
Stuor-oivi 68 40 58.40 8530517.90
Fuglenaes 70 40 11.23 9257921.06

From the are measured in Cape Colony by Sir Thomas Maclear in long. 18 deg. 30', we have

deg. ' " Ft.
North End 29 44 17.66 ..
Heerenlogement Berg 31 58 9.11 811507.7
Royal Observatory 33 56 3.20 1526386.8
Zwart Kop 34 13 32.13 1632583.3
Cape Point 34 21 6.26 1678375.7

And, finally, for the Peruvian arc, in long. 281 deg. 0',

deg. ' " Ft.
Tarqui 3 4 32.068 ..
Cotchesqui 0 2 31.387 1131036.3

Having now stated the data of the problem, we may seek that oblate ellipsoid (spheroid) which best represents the observations. Whatever the real figure may be, it is certain that if we suppose it an ellipsoid with three unequal axes, the arithmetical process will bring out an ellipsoid, which will agree better with all the observed latitudes than any spheroid would, therefore we do not _prove_ that it is an ellipsoid; to prove this, arcs of longitude would be required. The result for the spheroid may be expressed thus:--

a = 20926062 ft. = 6378206.4 metres.
b = 20855121 ft. = 6356583.8 metres.
b : a = 293.98 : 294.98.

As might be expected, the sum of the squares of the 40 latitude corrections, viz. 153.99, is greater in this figure than in that of three axes, where it amounts to 138.30. For this case, in the Indian arc the largest corrections are at Dodagunta, + 3.87", and at Kalianpur, - 3.68". In the Russian arc the largest corrections are + 3.76", at Tornea, and - 3.31", at Staro Nekrasovsk. Of the whole 40 corrections, 16 are under 1.0", 10 between 1.0" and 2.0", 10 between 2.0" and 3.0", and 4 over 3.0". The probable error of an observed latitude is +- 1.42"; for the spheroidal it would be very slightly larger. This quantity may be taken therefore as approximately the probable amount of local deflection.

If [rho] be the radius of curvature of the meridian in latitude [phi], [rho]' that perpendicular to the meridian, D the length of a degree of the meridian, D' the length of a degree of longitude, r the radius drawn from the centre of the earth, V the angle of the vertical with the radius-vector, then

Ft.
[rho] = 20890606.6 - 106411.5 cos 2[phi] + 225.8 cos 4[phi]
[rho]' = 20961607.3 - 35590.9 cos 2[phi] + 45.2 cos 4[phi]
D = 364609.87 - 1857.14 cos 2[phi] + 3.94 cos 4[phi]
D' = 365538.48 cos [phi] - 310.17 cos 3[phi] + 0.39 cos 5[phi]
Log r/a = 9.9992645 + .0007374 cos 2[phi] - .0000019 cos 4[phi]
V = 700.44" sin 2[phi] - 1.19" sin 4[phi].

A.R. Clarke has recalculated the elements of the ellipsoid of the earth; his values, derived in 1880, in which he utilized the measurements of parallel arcs in India, are particularly in practice. These values are:--

a = 20926202 ft. = 6378249 metres,
b = 20854895 ft. = 6356515 metres,
b : a = 292.465 : 293.465.

The calculation of the elements of the ellipsoid of rotation from
measurements of the curvature of arcs in any given azimuth by means of
geographical longitudes, latitudes and azimuths is indicated in the
article GEODESY; reference may be made to _Principal Triangulation_,
Helmert's _Geodasie_, and the publications of the Kgl. Preuss. Geod.
Inst.:--_Lotabweichungen_ (1886), and _Die europ. Laengengradmessung in
52 deg. Br._ (1893). For the calculation of an ellipsoid with three
unequal axes see _Comparison of Standards_, preface; and for
non-elliptical meridians, _Principal Triangulation_, p. 733.

_Gravitation-Measurements._

According to Clairault's theorem (see above) the ellipticity e of the mathematical surface of the earth is equal to the difference (5/2)m -ss, where m is the ratio of the centrifugal force at the equator to gravity at the equator, and ss is derived from the formula G = g(1 + ss sin squared[phi]). Since the beginning of the 19th century many efforts have been made to determine the constants of this formula, and numerous expeditions undertaken to investigate the intensity of gravity in different latitudes. If m be known, it is only necessary to determine ss for the evaluation of e; consequently it is unnecessary to determine G absolutely, for the relative values of G at two known latitudes suffice. Such relative measurements are easier and more exact than absolute ones. In some cases the ordinary thread pendulum, i.e. a spherical bob suspended by a wire, has been employed; but more often a rigid metal rod, bearing a weight and a knife-edge on which it may oscillate, has been adopted. The main point is the constancy of the pendulum. From the formula for the time of oscillation of the mathematically ideal pendulum, t = 2 [pi] [root](l/G), l being the length, it follows that for two points G1/G2 = t2 squared/t1 squared.

In 1808 J.B. Biot commenced his pendulum observations at several stations in western Europe; and in 1817-1825 Captain Louis de Freycinet and L.I. Duperrey prosecuted similar observations far into the southern hemisphere. Captain Henry Kater confined himself to British stations (1818-1819); Captain E. Sabine, from 1819 to 1829, observed similarly, with Kater's pendulum, at seventeen stations ranging from the West Indies to Greenland and Spitsbergen; and in 1824-1831, Captain Henry Foster (who met his death by drowning in Central America) experimented at sixteen stations; his observations were completed by Francis Baily in London. Of other workers in this field mention may be made of F.B. Luetke (1826-1829), a Russian rear-admiral, and Captains J.B. Basevi and W.T. Heaviside, who observed during 1865 to 1873 at Kew and at 29 Indian stations, particularly at More in the Himalayas at a height of 4696 metres. Of the earlier absolute determinations we may mention those of Biot, Kater, and Bessel at Paris, London and Koenigsberg respectively. The measurements were particularly difficult by reason of the length of the pendulums employed, these generally being second-pendulums over 1 metre long. In about 1880, Colonel Robert von Sterneck of Austria introduced the half-second pendulum, which permitted far quicker and more accurate work. The use of these pendulums spread in all countries, and the number of gravity stations consequently increased: in 1880 there were about 120, in 1900 there were about 1600, of which the greater number were in Europe. Sir E. Sabine[6] calculated the ellipticity to be 1/288.5, a value shown to be too high by Helmert, who in 1884, with the aid of 120 stations, gave the value 1/299.26,[7] and in 1901, with about 1400 stations, derived the value 1/298.3.[8] The reason for the excessive estimate of Sabine is that he did not take into account the systematic difference between the values of G for continents and islands; it was found that in consequence of the constitution of the earth's crust (Pratt) G is greater on small islands of the ocean than on continents by an amount which may approach to 0.3 cm. Moreover, stations in the neighbourhood of coasts shelving to deep seas have a surplus, but a little smaller. Consequently, Helmert conducted his calculations of 1901 for continents and coasts separately, and obtained G for the coasts 0.036 cm. greater than for the continents, while the value of ss remained the same. The mean value, reduced to continents, is

G = 978.03(1 + 0.005302 sin squared[phi] - 0.000007 sin squared 2[phi])cm/sec squared.

The small term involving sin squared 2[phi] could not be calculated with sufficient exactness from the observations, and is therefore taken from the theoretical views of Sir G.H. Darwin and E. Wiechert. For the constant g = 978.03 cm. another correction has been suggested (1906) by the absolute determinations made by F. Kuehnen and Ph. Furtwaengler at Potsdam.[9]

A report on the pendulum measurements of the 19th century has been
given by Helmert in the _Comptes rendus des seances de la 13^e
conference generale de l'Association Geod. Internationale a Paris_
(1900), ii. 139-385.

A difficulty presents itself in the case of the application of measurements of gravity to the determination of the figure of the earth by reason of the extrusion or standing out of the land-masses (continents, &c.) above the sea-level. The potential of gravity has a different mathematical expression outside the masses than inside. The difficulty is removed by assuming (with Sir G.G. Stokes) the vertical condensation of the masses on the sea-level, without its form being considerably altered (scarcely 1 metre radially). Further, the value of gravity (g) measured at the height H is corrected to sea-level by + 2gH/R, where R is the radius of the earth. Another correction, due to P. Bouguer, is -(3/2)g[delta]H/[rho]R, where [delta] is the density of the strata of height H, and [rho] the mean density of the earth. These two corrections are represented in "Bouguer's Rule": g_H = g_s(1 - 2H/R + 3[delta]H/2[rho]R), where g_H is the gravity at height H, and g_s the value at sea-level. This is supposed to take into account the attraction of the elevated strata or plateau; but, from the analytical method, this is not correct; it is also disadvantageous since, in general, the land-masses are compensated subterraneously, by reason of the isostasis of the earth's crust.

In 1849 Stokes showed that the normal elevations N of the geoid towards the ellipsoid are calculable from the deviations [Delta]g of the acceleration of gravity, i.e. the differences between the observed g and the value calculated from the normal G formula. The method assumes that gravity is measured on the earth's surface at a sufficient number of points, and that it is conformably reduced. In order to secure the convergence of the expansions in spherical harmonics, it is necessary to assume all masses outside a surface parallel to the surface of the sea at a depth of 21 km. (= R x ellipticity) to be condensed on this surface (Helmert, _Geod._ ii. 172). In addition to the reduction with 2gH/R, there still result small reductions with mountain chains and coasts, and somewhat larger ones for islands. The sea-surface generally varies but very little by this condensation. The elevation (N) of the geoid is then equal to _ /[pi] N = R | FG^(-1) [Delta]g_[psi] d[psi], _/

where [psi] is the spherical distance from the point N, and [Delta]g_[psi] denotes the mean value of [Delta]g for all points in the same distance [psi] around; F is a function of [psi], and has the following values:--

+-------+-------+
| [Psi]=| F= |
+-------+-------+
| 0 deg. | 1 |
| 10 deg. | 1.22 |
| 20 deg. | 0.94 |
| 30 deg. | 0.47 |
| 40 deg. | -0.06 |
| 50 deg. | -0.54 |
| 60 deg. | -0.90 |
| 70 deg. | -1.08 |
| 80 deg. | -1.08 |
| 90 deg. | -0.91 |
| 100 deg. | -0.62 |
| 110 deg. | -0.27 |
| 120 deg. | +0.08 |
| 130 deg. | 0.36 |
| 140 deg. | 0.53 |
| 150 deg. | 0.56 |
| 160 deg. | 0.46 |
| 170 deg. | 0.26 |
| 180 deg. | 0 |
+-------+-------+

H. Poincare (_Bull. Astr._, 1901, p. 5) has exhibited N by means of Lame's functions; in this case the condensation is effected on an ellipsoidal surface, which approximates to the geoid. This condensation is, in practice, the same as to the geoid itself.

If we imagine the outer land-masses to be condensed on the sea-level, and the inner masses (which, together with the outer masses, causes the deviation of the geoid from the ellipsoid) to be compensated in the sea-level by a disturbing stratum (which, according to Gauss, is possible), and if these masses of both kinds correspond at the point N to a stratum of thickness D and density [delta], then, according to Helmert (_Geod._ ii. 260) we have approximately

3 g /[delta]D \
[Delta]g = -- -- ( -------- - N ).
2 R \ [rho] /

Since N slowly varies empirically, it follows that in restricted regions (of a few 100 km. in diameter) [Delta]g is a measure of the variation of D. By applying the reduction of Bouguer to g, D is diminished by H and only gives the thickness of the ideal disturbing mass which corresponds to the perturbations due to subterranean masses. [Delta]g has positive values on coasts, small islands, and high and medium mountain chains, and occasionally in plains; while in valleys and at the foot of mountain ranges it is negative (up to 0.2 cm.). We conclude from this that the masses of smaller density existing under high mountain chains lie not only vertically underneath but also spread out sideways.

_The European Arc of Parallel in 52 deg. Lat._

Many measurements of degrees of longitudes along central parallels in Europe were projected and partly carried out as early as the first half of the 19th century; these, however, only became of importance after the introduction of the electric telegraph, through which calculations of astronomical longitudes obtained a much higher degree of accuracy. Of the greatest moment is the measurement near the parallel of 52 deg. lat., which extended from Valentia in Ireland to Orsk in the southern Ural mountains over 69 deg. long, (about 6750 km.). F.G.W. Struve, who is to be regarded as the father of the Russo-Scandinavian latitude-degree measurements, was the originator of this investigation. Having made the requisite arrangements with the governments in 1857, he transferred them to his son Otto, who, in 1860, secured the co-operation of England. A new connexion of England with the continent, via the English Channel, was accomplished in the next two years; whereas the requisite triangulations in Prussia and Russia extended over several decennaries. The number of longitude stations originally arranged for was 15; and the determinations of the differences in longitude were uniformly commenced by the Russian observers E.I. von Forsch, J.I. Zylinski, B. Tiele and others; Feaghmain (Valentia) being reserved for English observers. With the concluding calculation of these operations, newer determinations of differences of longitudes were also applicable, by which the number of stations was brought up to 29. Since local deflections of the plumb-line were suspected at Feaghmain, the most westerly station, the longitude (with respect to Greenwich) of the trigonometrical station Killorglin at the head of Dingle Bay was shortly afterwards determined.

The results (1891-1894) are given in volumes xlvii. and l. of the
memoirs (Zapiski) of the military topographical division of the
Russian general staff, volume li. contains a reconnexion of Orsk. The
observations made west of Warsaw are detailed in the _Die europ.
Laengengradmessung in 52 deg. Br._, i. and ii., 1893, 1896, published by
the Kgl. Preuss. Geod. Inst.

The following figures are quoted from Helmert's report "Die Groesse der Erde" (_Sitzb. d. Berl. Akad. d. Wiss._, 1906, p. 535):--

_Easterly Deviation of the Astronomical Zenith_.

Name. Longitude.
deg. ' "
Feaghmain -10 21 -3.3
Killorglin - 9 47 +2.8
Haverfordwest - 4 58 +1.6
Greenwich 0 0 +1.5
Rosendael-Nieuport + 2 35 -1.7
Bonn + 7 6 -4.4
Goettingen + 9 57 -2.4
Brocken +10 37 +2.3
Leipzig +12 23 +2.7
Rauenberg-Berlin +13 23 +1.7
Grossenhain +13 33 -2.9
Schneekoppe +15 45 +0.1
Springberg +16 37 +0.8
Breslau-Rosenthal +17 2 +3.5
Trockenberg +18 53 -0.5
Schoensee +18 54 -2.9
Mirov +19 18 +2.2
Warsaw +21 2 +1.9
Grodno +23 50 -2.8
Bobruisk +29 14 +0.5
Orel +36 4 +4.4
Lipetsk +39 36 +0.2
Saratov +46 3 +6.4
Samara +50 5 -2.6
Orenburg +55 7 +1.7
Orsk +58 34 -8.0

These deviations of the plumb-line correspond to an ellipsoid having an equatorial radius (a) of nearly 6,378,000 metres (prob. error +- 70 metres) and an ellipticity 1/299.15. The latter was taken for granted; it is nearly equal to the result from the gravity-measurements; the value for a then gives [Sigma][eta] squared a minimum (nearly). The astronomical values of the geographical longitudes (with regard to Greenwich) are assumed, according to the compensation of longitude differences carried out by van de Sande Bakhuyzen (_Comp. rend, des seances de la commission permanente de l'Association Geod. Internationale a Geneve, 1893, annexe A.I._). Recent determinations (Albrecht, _Astr. Nach._, 3993/4) have introduced only small alterations in the deviations, a being slightly increased.

Of considerable importance in the investigation of the great arc was the representation of the linear lengths found in different countries, in terms of the same unit. The necessity for this had previously occurred in the computation of the figure of the earth from latitude-degree-measurements. A.R. Clarke instituted an extensive series of comparisons at Southampton (see _Comparisons of Standards of Length of England, France, Belgium, Prussia, Russia, India and Australia, made at the Ordnance Survey Office, Southampton, 1866_, and a paper in the _Philosophical Transactions_ for 1873, by Lieut.-Col. A.R. Clarke, C.B., R.E., on the further comparisons of the standards of Austria, Spain, the United States, Cape of Good Hope and Russia) and found that 1 toise = 6.39453348 ft., 1 metre = 3.28086933 ft.

In 1875 a number of European states concluded the metre convention, and in 1877 an international weights-and-measures bureau was established at Breteuil. Until this time the metre was determined by the end-surfaces of a platinum rod (_metre des archives_); subsequently, rods of platinum-iridium, of cross-section H, were constructed, having engraved lines at both ends of the bridge, which determine the distance of a metre. There were thirty of the rods which gave as accurately as possible the length of the metre; and these were distributed among the different states (see WEIGHTS AND MEASURES). Careful comparisons with several standard toises showed that the metre was not exactly equal to 443,296 lines of the toise, but, in round numbers, 1/75000 of the length smaller. The metre according to the older relation is called the "legal metre," according to the new relation the "international metre." The values are (see _Europ. Laengengradmessung_, i. p. 230):--

Legal metre = 3.28086933 ft., International metre = 3.2808257 ft.