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OCR for page 167
C:EIAPTER XI
.. ..
GRAVITY MEASUREMENTS WITH THE LOT V()S TORSION
BALANCE ~
DONALD C. BARTON
Consulting Geologist and Geophysicist, Howslon, Texas
The essential working system of an Eotvos torsion balance consists
of' a vertical torsion wile suspended by its upper end and carrying a
horizontal aluminum bar susper~(lecl from its lower end. One end of' the
, F0 ~
at. _
_
A
Z
FIG. 1.
D'
bar carries a platinum or gold weight, and the other end carries a weight
of equal mass suspended by a fine wire some centimeters below the hori
zontal plane of the bar.
If a horizontal moment of torsion acts on the weights, the balance bar
will turn and will twist the torsion wire; the torsion wire will resist the
torsion and the balance bar will come to rest when the resistance of the
* The writer is indebted to Walter D. Lambert and F. E. NATright for reading and
helpfully criticizing the manuscript of this paper.
167
OCR for page 167
168
FI CURE HE THE EAR TH
torsion wire to torsion is equivalent to the torque of rotation. If' the
rotation of the wire is small, the resisting torque of the wire is directly
proportional to the angular displacement. The torque necessary to twist
the torsion wire through any given angle of displacement can be de
termined in the laboratory and a coefficient of torsion determined for
the wire. If the torsion constant of a wire is known and if an unknown
torque causes a rotation of the balance, the magnitude of the torque can
be determined by measuring the angle through which the balance has
been rotated by the torque. The angular displacement in practice is
measured by the device of a mirror mounted in the vertical axis of rota
tion of the balance bar and reflecting the image of a fixed scale to a fixed
telescope, or of a fixed beam of light to a fixed photographic plate.
In a gravitational field where the level surfaces (equipotential surfaces)
are parallel plane surfaces or parallel spherical surfaces there is no rota.
tion of the balance. If the balance is rotated, each weight rotates within
the level surface in which it was at the start. In such a gravitational
field, therefore, there can be no moment of torsion due to the effect of
gravity on the balance.
If the level surfaces are not parallel or if they are not either plane or
spherical surfaces, rotation of the balance will in general cause each
of the weights to cut the level surfaces. According to the law that any
body which is free to do so, will move (fall or slide) from a higher to
a lower level (equipotential) surface, the weights of the Eotvos torsion
balance will tend to rotate, falling horizontally from higher level surfaces
to lower level surfaces. In everyday life, level surfaces are, for all intents
and purposes, horizontal and parallel and bodies fall vertically or slide
down inclines. Normally, within the system of the Eotvos torsion balance
the level surfaces near each weight are not horizontal with respect to
the center of gravity of the balance, and as the only movement that is
possible for the weight is horizontal rotation, it tends to fall horizontally
through the inclined level surfaces.
The distortion of the level surfaces caused by the introduction of a
heavy mass such as that of Figure 2 into a gravitational held in which
the level surfaces are parallel surfaces, plane or spherical, is twofold. First
the level surfaces are crowded more closely together over the heavy mass
if the mass were lighter than normal, the level surfaces would be spread
apart; the level surfaces therefore are no longer parallel, and concomi
tantly the lines of the vertical are no longer straight lines but are curved;
within the system of the Eotvos torsion balance they approximate very
closely to small portions of circles of very large radius. Second, each
level surface normally is warped more in one direction than in another.
If the heavy mass is an infinite horizontal prism, the maximum distortion
OCR for page 167
GRAVITY MEASUREMENTS WI TH TORSION BALANCE 169
of the level surfaces is in the vertical plane at right angles to the longi
tudinal axis of the prism,'and there is no distortion parallel to the
longitudinal axis. If the prism is finite, there is also distortion in the
vertical planes parallel to the longitudinal axis; and in general, if the
heavy mass is of finite extent, it will produce a maximum distortion
of the level surfaces in one direction and a minimum in the direction
at right angles to it. If an Eotvos torsion balance is set free in a gravita.
18 ~
~ ~ ~ W~
~ \ ~ /
ma/ ~1
b \,/
Cow~
FIG. 2. Vertical section through ~ gravitationa1 field.
a, b tic = level surfaces, V1, A, V, V`< = lines of the ~ ertica1 TB\V~Y, =
Eotvos balance, TV1 = upper weight, W = lower weight.
tional field ire which the level surfaces within the held of' the balance
deviate in either or both of' these ways f:'rom parallel and plane or spherical
surfaces, the level surfaces in the vicinity of each weight will not be
horizontal, and the weights will tend to i'a11 horizontally through the
level suri:'aces and therefore to rotate, until the weights are in the lowest
level surl'ace touchecl. by their path of' swing, or until the Consent
causing then to rotate is counterbalanced by the resistance of' the torsion
wire.
OCR for page 167
1 7'0 PI G (JRE OF TIl E EAR TH
The situation in which there is merely differential warping of the level
surfaces is illustrated by Figure 2. The Eotvos torsion balance in that.
case acts the same as the much older, or Coulomb, type of torsion balance,
.~
/
i
/~
i/
a an_
. L I _
ok
\:
A\
.~
:^C~^
:'
/1
?C
~ j ~ , 'G
FIG. 3. Blocl: cliagram of space around an Eotvos or Coulomb b;llance ~epre
sentin~gr, curvature of the level surfaces.
in which both weights are in the horizontal axis ot the bar and therefore
in the same horizontal plane. Figure 3 represents the block diagrams
of the space of a Coulomb balance, or ot all Mottos torsion balance;
AEF and COG represent, respectively, the vertical planes ot maximun~
OCR for page 167
~]P7~F ~S ~ r0~\ ~ 171
and minimum ping ABC represents ~ borizont~Ipl~ne~ KLIII
TepTeseDtatbeVertic~lcyliniertb~oughtbep~thoTs~ingoItbevei~t
W;~,c,],representtbetr~cesoflevelsud~cesin Deplanes AFf
aniCB08~]intbecylinierELlII;~TIgIbreprosentstbetIRceof
tbeborizont~Ipl~ein ~bichtbe weights move;I ~] IIrepresent
posgionsofibeveight W. Ibe~eight W intbepos~ionsI~n] 1I ~
indi~erentlevelsurfaces, BDi, OS tbe~i~gr~misir~wn~inpositionI
the vei~tisin the biggest India position IIin thelo~estleveI
sUrf8Ceinto~bichitsp~tboisvingc~r~Tesit.IftbereweTenoresistanCe
to its rotation, the weight W would I'll borizont~lly from position I; or
Jrom any pos~lon Between positions I ~d II, into position II ~] come
to rest there. Si~Tl~rly for the other quadrants, the weight voUl] ten]
to I'll horizontally into and come to rest in the nearer of the two points;
either II or its corresponding point on the hack of the diagram; the
tendency to angular flotation is the same for totb Feints of the balance.
Tbo situation Obese We level sedges are not parallel is illUstl~teJ
by Figure 4. Ibe hgUre gun represents ~ block diagram of the space
around an Edtv~s torsion balance. ALFt and SCOW represent ~spec
tively, vertic~1 pleaded of m~xT~um divergence RD] of parallelism of the
level sUrI~ces; a; t, c; d; e, represent the traces of level surfaces; We
level surface c Coincides with the borizont~1 plane thfongh the center
of grain of go torsion Balance system; the upper Feint LW ~D] We
lover weight [W rotate in parallel bo~zont~1 planes [~\ ~] O[Q.
If the Weight OW is in And position but II the level surI~ccs in its
vicinity chili he inclined to its pub of swing, and if there were go re
sist~Dce to its rotation it would Suing into ~] come to rest in position
IT, in ~icb position it is in the lowest level surface Ibid its path of
suing cuts. If the Feint [W is in ~ position other tab its position
in Me bidden I've AB~ of the Cloak, it Fig I~I1 bori%ont~lly tbroUgb
the lean surfaces and come to rest in that position, in Ibid it is the
ingest level surface abut its pub of suing CUtS. Again; the tendency to
angular rotation is the same for totb weights of We balance.
The ho~zont~1 rotations or bo~zoDt~1 Idlings of the weights tbroUgb
alla level surfaces is of Ours JUe to We fact ~ inclination of the
level surfaces to tab p~ of suing of the weights necessarily is ~c
co~p~nie] by inchn~tioD of the vertic~1 to the Tub of swing of the
wei~tsan]tb~t ~ereIoro Gavin b~s~sm~llborizontaTcompoDent
~ron~lennib~eT~t~ Them~ni~de~ ~#hnr~n~t~rn~
~ . , . ~ . ~
poneutoIgr~vityr~gLeseenIrom theprecediDg~iscussionstote~
function oft~oiniepenJe~tI~ctor~ tbeoncthcdi~ereDti~ warping
oIthAleYPlEurf~ces>unitheothbrtbedivergeDccoftbeleYclsUrf~cos)
orvL~tisthos~methi~g,thccurv~tureof~l~vertToil. from thcin
OCR for page 167
172 FIGURE OF THE EARTH
spection of the blocl; diagrams, it is of course selfevident, that the magn;
tude of that horizontal component of gravity at any point in the path
of swing of the weights is a function also of' the azimuth of that, point.
/
OCR for page 167
GRAVITY MEASUREMENTS WI TH TORSION BALANCE 173
difference between the radii of curvature of the level surfaces in the
vertical planes of the maximum and minimum warping; i. e.:
q x ~ R
min
r' ~ xf<~y,
imax
(I)
where IN is the mass of a weight times the square of the lever arm of
the balance.
~ level surface taken over any considerable area is a very irregularly
warped surface and cannot be represented by annoy simple mathematical
surface. But within the very small area of an Eotvos or Coulomb balance,
the level surfaces approximate closely to surfaces all of whose vertical
profiles parallel to the planes of maximum and minimum curvature are
small portions of very large circles of either positive or negative curvature.
Such surfaces may be small portions of synclastic surfaces, which are
small portions of very large ellipsoidal, paraboloidal, hyperboloidal or
similar s.ynclastic surfaces; or saddleshaped, anticlastic surfaces. The
discussions that follow are illustrated by the case in which the level
surfaces are very small portions of a very large ellipsoiclal or similar
s~Tnclastic surface, but the derivation of the formulae is algebraic and is
equally valid for the cases where the curvature in one of the principal
directions is up (positive) and in the other down (negative) or where
the curvature in both of the principal directions is up. The terms " maxi
mum and minimum curvature," furthermore, do not refer to the geo
metrical maximum and mini nun curvature but to the algebraical ma:;i
~num and minimum curvature.
In Figure 5, let:
i be the trace of a level surface in the horizontal plane ABCD:
DCGF be the perpendicular plane parallel to the major axes, MM' of
that level surface; and
ADFE be the corresponding plane parallel to the minor axes ON';
k and h be the respective traces of that level surface in DCGF and ADFE.
BD be the projection on ABCD of the balance beam;
I be the half length of the balance beam.
AN: be one of the weights of the balance and be in the level surface, hilt
a be the angle of azimuth of the balance beam.
= mass of the weight;
rat and ret be the radii of h and k, respectively.
go and go be the projections respectively on ADFE and DCGF of an
arrow representing~the value of gravity at the position of NV. As
the angle between go (or it.,) and the vertical is extremely small,
and as g1 (or q~g times the cosine of an extremely small angle,
I, and I., may be tolled as equal to I.
.,
OCR for page 167
~4
FI G URE OF THE EAR TH
From the relation of the vertical to a level surface g1 and go are perpen
dicular to h and k, respectively, and therefore are no longer perpendicular
B
I've
/
1
~. ,
/ 'it ~ '1
~,,''~/'\q
>
in/
G
l ~ ~ ~G
mu/
\
F
FIG. 5.
/
to the hori%or~ta1 plane, and each has a horizontal co~npollent, S and T.
respectively.
OCR for page 167
GYRAVI TY MEASUI7EMENTS WI TH TORSIOA! BALANCE 175
Rememberin', the assumptioI1 that th.e profile o:t the level surface is a
part of a very large circle:
S =: g COS ~ = ,7 = q ~ I Sill a
T = g cos ~ = ~t = c7 1 1 cos a
~ r

B
A<
1
\
/~
~ ,'~`
\^'~' \ '
~," \
~S \
`80°4 ~ \
~ ~ ~ _
a ~ b
FIG. 6.
f
/~
,~180°~
. ~.
(2)
(3)
> G
Let Figure 6 represe~:~t tI~e hori.zontal plane through the position of W.
aW and bW then will be the components of S and T. respect.ively, acting
perpendicularly to the balan.ce beam ancl their difference multiplied b~
the distance I and the mass will be the torque F. tending to rotate the
balance beam.
aW=S cos (180°a~gsin~ cos a (4)
bW=T sir1 ( 180°a~ =g ~ cos a si~i a
12
F=(rnl)(bV~:aTV)=(rttl)(~( ~  ~ Jisin2a (5)
OCR for page 167
FIGURE OF THE EAR TH
()is a constant for any given level surface ancl ~ ( ~ )
is the " curvature " value of Mottos, often called for short " R " or " R
value" from the German word "Richtkraft" (directive force).:
The force corresponding to the torque ~ is represented geometrically
by of. The line eW represents the projection of (g) on the horizontal
surface and ef is the component of eW that is perpendicular to the.
balance beam at the weight. If the level surface is a plane, eW becomes
~ point and ef vanishes. If the level surface is spherical, em lies in.
the balance beam and of again vanishes.
If the axes of reference do not coincide with the major and minor axes
of the curvature, and if A its the angle between the two sets of axes and
if ,8 is the azimuth of the balance beam in reference to the axes of
reference,
a=A,0
1~ 1~
k7=ra 2 R sin 2a=m 2 R sin 2(A,3)
=m 2 R sin IN cos 28to 2 R cos 2A sin 23 (6)
FOR sin 2A) and (R cos 2A) are constants for ally given level surface
and for any given set of axes.
By differential geometry it is possible to show that:
OR sin 2A=~ andR cos 2A= ~
~Obey By'' Dx
:From those relations:
or
or
(a)
Ret sins 2A+R~ costs 2~=4Uxy~+ Use 2
V4U ~ + U ~ _ R sin 2A _
in' R cos 2A
tan 2A = _ 2 US
2 U71'
or

(a)
* It is obviously not of the right dimensions to be a force; its dimensions are
those of the gradient of an acceleration.
The terms a U , a U. a U and U U often shortened to Ups Ups
DxAz ayaz Obey' By Ox~ '
US and USUXX or Ups, are the shorthand symbols of calculus for these and
two other quantities, to be discussed shortly, U being the Newtonian gravitational
potential. Any one approaching the theory from the geometrical side and not
understanding the calculus of the potential function has to accept them merely as
arbitrary symbols.
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GRAVITY MEASUI?EMENTS WI TH TORSION BALANCE 177
If at least two independent values of F have been obtained by observa.
tion at any point with ,8 chosen at will and therefore known, the equation
can be solved for US and Up . The magnitude and azimuth of the Eotvos
curvature value can then be calculated from those quantities by the
I'ormul~e of equation ( 8 ~ .
The Eotvos torsion balance is a special type of the Coulomb balance
if' the assumption is made that within the very small area of the working
system of' the Lotvos torsion balance the curvature of' all level surfaces
is identical. The upper level surl'a,ces in general will tell to have slightly
flatter curvature there the lower level sur:t'aces. But the ma';in~um differ
er.~ce in the curvature of:' the higher and the lower level surfaces is very
small compared with the difference between the curvature in the plane
of' the major and that in the plane of the minor axes except when the
curvature~of the level surfaces is very nearly the same in all directions.
The vertical difference in the curvature therefore can be neglected. It'
the curvature is the same at all levels of a system, the horizontal directive
force is the same at similar points at all levels, and the force acting on
the weights of a Coulomb balance is the same whether the weights. are at
the ends of the beam as in the original Coulomb balance or whether
one of the weights is suspended perpendicularly below the end of the
beam as in the Eotvos torsion balance.
Baron Eotvos' great contribution was in demonstrating that if one of'
the weights of the balance is below the plane of the other, the hori%ont.al
component of gravity afI:'ecting the balance is a function also of a magni
tude that depends. on the curvature of the vertical and that the latter
depends on the horizontal gradient of gravity.
In :Figure ~ let:
an' be the level surface through the center of gravity, of an Mottos
torsion balance;
Arc' and ac da' be vertical squares, i. e., N centimeters on a side, ~ being
small; V be the line of' vertical; pi be the value of ~ra.vity at a.
The difference between the value of gravity at a and at a' by deLni
t.ion will be N times the gravity ~,ra.dient, G~(g), and g+N Gr(~g)
will be the value of gravity at a'.
The vertical will be perpendicular to pa', but due to its curvature will
make an angle ~ a with be and ed.
The work done if a unit mass is moved from d to a' is the same whether
the path of the movement is via daa' or dea'' and will be
from Otto a' Nrg+NGr~g)] ~ from e to a Ng
from a to a' O  from d to e Ngcosa (9)
* This angle a is not connected with the azimuthal angle previously denoted by
the same symbol.
OCR for page 167
180
FI CURE OF THE EAR TH
The gradient and its azimuth are given by the relation
Gr~g)=~/U~.~+U'J,,,2'
U.~z
tan a = U
CALCULATION OF THE CURVATURE AND GRADIENT
(13)
The balance beam of the Mottos torsion balance through its weights is
under the influence of the two turning moments, one the horizontal
directive force, and the other the horizontal component of gravity vary
ing as the gradient. Each has a maximum magnitude and a direction
of that maximum magnitude. But each is independent of the other.*
In any particular position of the beam of the torsion balance, the magni
tude of each moment affecting the balance will not be the maximum
value but some function of that maximum value and of the angle that
the balance beam makes with azimuth of the gradient or of the curvature.
The total moment of torsion, F. on the balance beam in any particular
position is the sum of the two and is expressed by the following formula:
~ = E US si n. 2a + K U9.1J cOs 2a1H U~.~ sin ~ + M Us cos a ~ 14 ~
Equation ( 6 ) plus equation ( 11 ) with substitution of equations ( 7 ) in
equation ~ 6 ~ .
Where:
hi' is an i~stru~llenta.l col:~sta~t depe~clillg on the gauss of tile weights,
the square of the half length of the balance bar; and
M is an ins.tru~ental constant depending on the Nash of the weights,
the hall' length of'; the balance bar, the. vertical clistance between the
weights.
The torque, ~ causes an angular rotation of' the balance s.ysten~ until
~ounte~.ha.lancecl by tilde :~esistanee of the torsion wire. The angle through
which the beans turns is deter~ni~:~ed by reading the deflection o:t' the
image o:t' a :fixed point in a mirror ol1 the axial stem of' the balance beam.
If' the angular: torsion constant of' the torsion wire has previously been
determinecl in the laboratory, it is then possible to calculate the magni
tude of the force causing the rotation.
If' four values for ~ are determined by observation, the equations can
then be solved for the values of' the four unknowns, (1~, Use,, Up, anal
L1~,~. From the nature of the torsion balance, the zero point of' the instru
ment enters the equations as a fifth unknown in the case of the obsolete
singlebalance type of' instrument and then at least five determinations
*Actually a complicated mutual relation exists.
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GRAVITY MEASUREMENTS WI TH TORSION BALANCE 181
of ~ are necessary. As the snorers doublebalance type of instrument
consists of two complete independent balance systems mounted closely
side by side but with a difference of orientation of 180°, the zero point
ofeach system enters the equation as an unknown and therefore with the
modern instrument at least six independent determinations of F are
necessary. As two determinations of if, one by each system in the instru
ment, are made in each position, the observation of the modern instrument
in three positions is sufficient mathematically for the determination of
all six variables, but as a matter of fact, at least four and usually five
positions are observed, the last two being a repetition of the first two
for the purpose of a check on and increase in the accuracy of the
observations.
In practical work, equation (14) is replaced by the equation
trtroof = K Us sin 2a+KUx.~ cOs 2aMUxz sin a+MUy~ cos at (~)
where nO is the zero point of the balance, rib the scale reading at any par
ticular observation, it'= T ~ and llI= TV ~ D is the distance from
mirror to scale in scale divisions, and T the torsion constant of the wire.
The value of it is observed at the constant set of azimuths, a= 0°, a= 120°'
and a=240° (more rarely at azimuths of a=0°' a=180°' a=9~)°'
a= 270° ). The solution of the resulting equations has been carried
algebraically and is now cast in the form of a simple tablelil~e form, in
which the observer has merely to enter his observed readings of the scale
in the proper spaces and then follow a simple indicated routine of ele
menta.ry addition, subtraction, multiplication and division to obtain
lye., Us,, U.,, Us', (if the 00° positions have been used, Us and Us,,
but not U.` or Ul'J). The respective values of the
182 FIGURE HE THE EARTH
close to the instrument cannot be corrected for, but they can be avoided
· ~ ~ · (' ~ 1 1 1 · · L · _ _ 11_ ~ 11~
in some cases by proper chose of the Stratton s~t.e; In other eases, they can
be eliminated to ~ considerable extent by taking additional stations
within a short distance of' the station, and in a great many areas,, they
are negligible.
The radius of' curvature of the curved vertical can be calculated if the
vn.l,~n nt' the gradient and of gravity are known. The diagram of Figure
, ~ · ,.
8 represents an upper unit teentlmeter) square or 11 lgure b; ~ represents
the radius of curvature at a and r+c represents a, radius of eurva.t.ure
prolo~`,,ed to the point b; a is the angle between the perpendicular anal
b
;~
c
a
Q'
r
FIG. 8.
the vertical assumed as ~ straight line; ,B is the angle between r and Or+.
a HAS, Gr (A ) = g sin a, and sin h = + or as ~ is e,xtren~elv pinball and r
very larger
sin ,0=, an<1 G? (g) =g.
(14)
The radius of' curvature of the curved vertical therefore equals the value
of' ~,ravit,y cl.ivided by the value of the ~ tidiest and is in terms of eenti
n~eters. For most practical purposes, the ~eciproeal of' the numerical
value of' the gradient ill Eotvos units is a, sufficiently close a,pproxin~ation
to the radius of curvature of' the vertical in terms of units of ~ x 10i'' em..
and can be obta;.necl without l
GRAVITY ^71lEASUREMENTS WITH TORSION BALANCE 183
the deviation of the principal axes of' the equipot,ential surface from the
axes of reference. As the torsion balance measures only the last, of those
quantities and the difference between the first two of' the quantities, the
form of the equipotential surface cannot be determined from torsion
balance observations alone. There is, however, an additional relation
het,ween the first three quantities;
alum + 82U + You _9,.,
a*'' BY''
~ ~~v, ~
where a, is tile angular velocity of rotation of' the earth. If a U is deter
minecl independently,, as for example by a Jolly balance,, it, is then
possible to calculate the value of' radii of' curvature of' the equipotential
surface. The accuracy of' the results is not nearly of the same high
magnitude as the quantities observed directly by the torsion balance, as
the accuracy of:' the Jolly balance and the other instruments for measuring
the quantity a U is not nearly comparable to that of' the torsion balance.
The i' orm of' the e quip otent.i al surface ancl the do fleets on of' the verti cat
can be calculated, however, i'or all stations of' a torsion balance survey, ii'
astronomicgeoclet,ic, measurements of' the deviation of' the vertical have
been made for two points with the accompanying determination of' the
n~ean curvature of' the level surface between the two points, if' the
torsion balance stations are closely spac,ecl, and if the terrain is nearly
level. These calculations were worked out by Ec~tvcis~ for his survey with
the torsion balance near Aracl in the Hungarian plain in an area now
just within the Roun~anian border but are not usually Treacle in connection
with torsion balance survey s.
The co~s.tancy of the t,ravit.atiorlal constant was cleterminecl by Eotvos,
through the use of the torsion balance, with an accuracy of better than
one part ice twenty nonillion hi contrast to the accuracy of' one part in
sixty thousand in the classical pendulum observations. by Bessel. The
first part ol' that series of:' Baron Ecitvos observations consisted of' a
study of:' the effects of' gravitation on cli~'erent substances, which were
used to replace the ortl;na.ry old or platinum weights of' his balances.
The st.ucly included not only the effects of the earth's attraction but also
the effects of the noon's and of' the sun's attraction. The second part
of' the observat.io~s consisted of' a study of any possible effect on the
attra.ctio~ on those substances if they were subjected to the influence of
radioactive preparations. The third part of the observations was a study
to see if' there was any " absorption " of' the earth's attraction by thick
lead plates introclucerl. around the weights of the torsion balance.
OCR for page 167
184
FIGURE OF THE EARTH
UNITS OF MEASUREMENT
The Newtonian gravitational potential, U. measured
in the cgs.
system in ergs per gram has the three first derivatives, a ~ it a
which give the rate of its variation respectively in the vertical and the
two horizontal directions, which are in fact accelerations, and which,
in the cgs. system, are expressible in dynes per gram. The rate of va.ri
ation of these three quantities in the two horizontal directions is given
respectively by
v
(1) Sax (0aU)'
and in the vertical by
(~)axtaz),
GRAVITY MEASUREMENTS WITH TORSION BALANCE 185
directions, R actually does not give that difference but gravity times it,
i. e.. R= a ( ~  ~ ), and strictly, it would be more correct to use the
~ max
\rmin
n,~nntit.~r R which would give the differential curvature (~
.... ~
\rmin rmaX J
The quantity,, would be expressible in units of radians per centi
meter (r. pi. c.), i. e., ~ cm. of arc subtending so many radians. But as
the range of the actual value of the reciprocal of the value of gravity
where torsion balance surveys are likely to be made is less than fivetenths
of one per cent, as the value of the reciprocal is about 1.02 x 103, and
as the value of R as observed in the field rarely is significant to within
five per cent., the numerical value of R in terms of Eotvos units is the
same for most practical purposes as the numerical value of the differential
curvature in terms of ~XiO~0 I. p. c. The curvature of the vertical
which is given by the quantity, (g), likewise is expressible in terms
of fix 10~ I. p. c., the numerical value of which will be the same for
most practical purposes as the numerical value of the gradient in Eotvos
units.
. .. .. ~,. · .
VARIATION OF GRAVITY
L7
Depending on the particular situation, stations can usually be placed
close enough together so that a straightline variation of the gradient.
may be assumed between adjacent stations. I]: JIG is the difference in
the ~T~liln nt ~r~vit~v in a. level surface and S the distance between two
adjacent stations and (I/ (g ~ a anal Gr (g ~ Z. the gradient respectively at
the two stations, then
~ '_ , ~ . .
~G=SGr (.~+ G7~(q)',
.
(14)
The stations in some situations must be only a few tens of meters apart
and in other situations may be several kilometers apart. If a net of
stations is thrown over an area with the mesh of the net close enough
so that a linear variation of the gradient between adjacent stations may
be assumed, a complete picture can be obtained of the horizontal variation
of gravity over the area. A triangular traverse some 120 miles in length
was run under the writer's direction and closed with an error of less
than 5 x 1O'3 dynes. The variation of gravity as calculated from such
a traverse gives an approximate picture of the variation of gravity in
a single level surface. If the traverse is tied into a point at which the
value of gravity is known and if the elevation of all the stations of the
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~ ~6
FI CURE OF THE EAR TH
traverse with reference to that point is known, a close approximation
to the actual value of' gravity at each station of' the traverse can be
determined.. Oltay 34 has shown that the value of' relative gravity deter
mined i'rom surveys with a torsion balance call be regarded as being as
accurate as the value of' relative gravity determined faith the i.nvariahle
pendulum provided that torsion balance stations are sufficiently close
together and the terrane is. rather flat. Oltav obtained values for relative
',ravity by the relative pendulum method at six key stations in the
area covered by a torsion balance survey made bar Baron E`.;tvos in the
Ara.d district. The differences between the torsion balance ancl the in
variable pendulum. determinations of' relative <,ravity at those stations
were respectively, 0.001, 0.001, 0.004, 0.000, 0.00d, and 0.000. The respec
tive distances between the pendulum stations were, i?, 9, 13, 20, 42,
ancl o0 kilometers.
The deviations from a still, homogeneous sphere of' the form and distri
hution of mass in the earth that appreciably affect the gradient and curva
ture values and gravity at the surface May be class;.fiecl in the i'ollowirlg
orders of' magnitude: first order, the planetary or " normal " deviation
clue to the rotation and the consequent oblate spheroidal i'orm of' the
earth; second order, the continental irregularities.; third order, the
irregularities due to large mountain systems, such as the Alps or Rocl:.~
Mountains; fourth order, the irregularities due to small mountain masses,
such as the Panhanclle graIlit.e ridge of' Texas; fifth order, the irregulari
ties clue to salt comes, volcanic necks, ancl such mocleratesized teleologic
structures; sixth order, irregularities due to very local structures such as
blurted stream channels, small faults; seventh order, survival irregu
larities ifs the soil and subsoil; and eighth or: filer, the topographic
i rregula.rities..
The to~'ogral~hie irregularities are ~ orrecterl for, atoll the surficial
;r regularities are avoided by judicious ch.o;ce of' station sites, if possible
or t}leir ell'ects are minilll;%e(l. be' acl(litioIla.l stations. The observe(l
.,
or
GRAI ITY MEASUREMENTS WITH TORSION BALANCE 1S,
leas been ini'errecl with a fairly Cool degree of' accuracy from the results
of' surveys with the Eotvos torsion balance. In a. study of' tonne eleven
hundred odd stations I'ro~n the Gull' Coast salt clomes, the Powell fault
Force area., acid southern Oklahoma, the writer plotted a graph of' the
i'requen.cv of' the difI:'ere~~t values of' the ma.~,:r~et.ic northsouth and east
~vest components of' the gradient at each station. By the use of' the mean
magnetic clecli~ation of the area in~olvecl to transform 'row magnetic:
to astronomic north and by the use of' the median value 'or each com
ponent, a value ol' 7.4 E (100 clynes) north about 4 cle$ree east was
obtained for the median value of' the observed gradients at the eleven
hu~drecl odd sta.t.ions. The theoretical values for the normal northward
,raclient were calculated from Helmert's formula and a mean value,
weighted according to the number of' stations in each latitude, was
obtained for the eleven hundred odd stations. The weighted theoretical
Clean value was T.14 E (due North of' course) in comparison with the
observed median value of' :.l E, N ~° :~. The observations at these eleven
hundred odd stations were made in commercial work in which, in
general, speed is considered more important than extreme scientific
accuracy, and in a large part of which the purpose of' the surveys was
qualitative or only roughly quantitative rather than accurately quanti
tative. The surveys furthermore were mostly around and on salt domes,
c; ~
~. ... ..
I'aults, and a.nticlines where the gra.cdients are abnormally large and varied
in direction.
~ moderately close parallelism commonly holds between the irregulari
ties of distribution of' mass and geologic structure. The older sediments
tend to be denser than the younger, the granite, gneiss, and schist and
Cambria.~Ordovici.an basement of many sedimentary basins tend to be
much denser than the overlying prism of sands and shales. Positive
rock masses, whether the upthrow side of' a fault, hoists, or anticlines,
tend in `~eneral.to be denser than the normal rocks at the same level.
NYhere thick salt masses form the core of' anticlines or of' salt domes,
I'ormations older than late Tertiary, the opposite holds true and the posi
tive mass as a whole may tend to be lighter than normal rocl~s at the
same level. Positive rock masses in general tend to be associated with
a gravity maximum, a gravity gradient toward the positive mass, and a
certain pattern of' curvature values, but ifs a few cases the positive mass
is associated with a gravity minimum, a ~ravit,v gradient away :t'rom the
positive mass, and an opposite pattern of:' the curvature values. Regional
and areal torsionbalance surveys can be used to map most. of' the major
and many of' the lesser structural features of' a region or area. The torsion
balance has been used with distinct success to map the buried gra.nite
g~eissschistOrdovician limestone ridges o:t' North Texas, the Texas
OCR for page 167
188
FIGURE OF THE EARTH
Panhandle, and Oklahoma, and has shown its competence to map molly
lesser structures. It has to its credit in the Gulf Coast of Texas the
discovery of the Nash, Allen, Clemens, and Lone, Point Shallow salt
domes, the Dewalt (deep) salt dome, the Esperson, :Hanka.~ner, awl
MyLawa (very deep ~ salt d omes, and many pi: ospec t:i ve al om.es. =~¢ ~ 
rate calculations of: the :form,.depth, and pos;.tior~ of the structure car.,
be made in some cases. The form and thicl:~ess of the caprock and the
depth to it and the salt can lie calculated Title lair accuracy for mall.:
of the Gulf Coast salt domes and with. high accuracy for a few. But
the to~sionbalance results in most cases ¢,i.ve only a qualitative picture
of the geologic structure.
REFERENCES
The original and still the most fundamental papers on the Eolvos torsion bal
ance are Baron Eotvos' original papers:
1. Eotvos, Roland von. Untersuchungen uber Gravitation und Erdmagnetismus.
Ann. Physik und Chem. n. I., Leipzig, Band 59 (1896).
2. . Bestimmung der Gradienten der Schwerkraft und ihrer Niveauflachen
mit Hulfe der Drehwage, Verh. der XV. allgemeinen Konferenz der
Internationalen Erdmessung in Budapest 1906, Bd. I: 337395.
. Die Niveauflachen und die Gradienten der Schwerkraft am Balaton
See. Budapest, 1908.
4. . Bericht fiber geodatische Arbeiten in Hungarn, besonders uber Beo
bachtungen mit der Drehwage, Verh. der XVI allegemeinen Konferenz
der Internationalen Erdmessung, London and Cambridge, 1909.
5. ., Pekar, D., and Fekete, E. Beitrage zum Gesetze der Proportionalitat
von Gravitat und Tragheit. Ann. Physik, IV, Bd. 68 (19Q2).
The more important papers in English are:
6. Rybar, Stephen. The Eotvos torsion balance and its application to the find
ing of mineral deposits. Econ. Geol. vol 18, no. 7: 639682 (1923).
7. Shaw, H., and LancasterJones, E. The Eotvos torsion balance. Phys. Soc.
London, Proc. vol 35, pt. 3: 151165 (1923).
8. ., and . The application of the Eotvos torsion balance to the in
vestigation of local gravitational fields. Phys. Soc. London, Proc. vol. 36,
pt. 4 (1923).
9. ., and . The Eotvos torsion balance and its application to the
location of minerals. Mining Mag., London (Jn.Fb.) 1925: 1826, 8693.
10. Fordham, W. Herbert. Oilfindina by geophysical methods. J. Inst. Petroleum
Tech., London? no. 52, (Oct., 1925); reprinted and distributed by L. Oert
ling, Ltd., London.
11. Anonymous (probably H. Shaw and E. LancasterJones or some pupil of
theirs). The Eotvos torsion balance. London, L. Oertling, Ltd., 1925. (To
be used as a manual with the Oertling torsion balance.)
12. Ambronn, Richard. Methoden der angewandten Geophysik. Leipzig, 1926.
English trans. by Margaret C. Cobb as follows: Elements of geophysics,
as applied to explorations for minerals. oil and gas. New York, McGraw
Hill Book Co., 1928.
OCR for page 167
GRAVITY MEASUREMENTS WITH TORSION BALANCE 180
13. Shaw, H., and LancasterJones E. Locating minerals and petroleum. Mining
Mag., London (Ap.Ml. 1027). A series of articles reprinted and distributed
by L. Oertling, Ltd., London, 1927.
14. Barton, Donald C. The Eotvos torsion balance method of mapping geologic
structure. Amer. Inst. Mining Met. Eng., Tech. Publ. no. 50 ( 1928),
also in Geophysical Prospecting 1929, Amer. Inst. Mining Met. Eng. (1929).
15. . Calculations in the interpretation of observations with the Eotvos
torsion balance. Geophysical Prospecting 1929, Am. Inst. Mining Met.
Eng. (1929).
16. LancasterJones, E. Computation of Eotvos gravity effects. Geophysical
Prospecting 1929, Am. Inst. Mining Met. Eng. (1929).
17. Shaw, H. Gravity surveying in Great Britain. Geophysical Prospecting 1929,
Am. Inst. Mining Met. Eng. (1929).
18. Heiland, C. A. A cartographic correction for the Eotvos torsion balance. Geo
physical Prospecting 1929, Am. Inst. Mining and Met. Eng. (1929).
19. . A new graphical method for torsion balance topographic corrections
and interpretations. Bull., Am. Assoc. Petrol. Geol., vol. 13, no. 1: 3974
(1929) .
20. George, P. W. Experiments with Eotvos torsion balance in the TriState zinc
and lead district. Geophysical Prospecting 1929, Am. Inst. Mining
Met. Eng. (1929).
21. Barton, Donald C. Tables of terrane effects. Bull., Am. Assoc. Petrol. Geol.,
vol. 13, no. 7: 763778 (1929) .
22. . Control and adjustment of surveys with magnetometer or the torsion
balance. Bull., Am. Assoc. Petrol. Geol., vol. 13, no. 9: 11631186 (1929).
23. Shaw, H. Interpretation of gravitational anomalies. Am. Inst. Mining Met.
Eng., Tech. Pub. 178 (1929).
24. . Interpretation of gravitational anomalies II. Am. Inst. Mining Met.
Eng. In press.
25. MeLintock, W. F. P., and Phemister, J. The use of the torsion balance in
the investigation of geologic structure in Southwest Persia. Summary of
the progress of the Geological Survey of Great Britain for the year 1926.
H. M. Stationery Office (1927).
26. LancasterJones, E. The computation of gravitational effects due to irregular
mass distributions. Monthly Notices Roy. Astron. Soc., Geophys. Suppl.,
vol. II, no. 3 (May 1929).
27. Barton, Donald C. The torsion balance in the determination of the figure
of the earth. Am. J. Sci., 18: 149154 (Aug. 1929~).
28. Heiland, C. A. Geophysical methods of prospecting. Principles and recent
successes. Colorado School of Mines Quarterly, vol. 24, no. 1: 1747
(March 1929).
29. McLintock, W. :F. P., and Phemister, James.
I. A gravitational survey over the buried Kelvin Valley at Drumry near
Glasgow, and
II. A gravitational survey over the Pentland Fault, near Porto Bello, Mid
lothian, Scotland. Summary of the progress of the Geological Survey for
1928, Part II: 128. H. PI. Stationery Office. London, 1929.
30. and . A gravitational survey over the Swynnerton Dike, Yarn
field, Staffs. Summary of progress of the Geological Survey for 1927. p.
114. H. M. Stationery Office. 1928.
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1.'30
FI CURE OF THE EAR TH
31. McLintock Hi. F. P. and Phen~ister, James. A gravitational survey over the
buried Kelvin Valley at Drumry, neat Glasgow. Trans. Roy. Soc., Edin
burgh 66: 147 (1929).
32. Jones, J. H., and Davies, R. The measurement of the second derivatives of
the gravitational potential over a buried anticline. Monthly Notices,
Roy. Astro. Soc., Geophys. Suppl. v ol. II, no. 1: 132 (1928).
33. Numerous, B. Gravity observations in the Solikamsk and Beresniaky Districts
in the northern Urals in 1926 and 1927.
Results of gravity observations of 1928 near Lake Ba~kunchak,
Results of the general gravity survey in the Emba District,
Results of gravitational observations in the region of Grosny in 1928,
Results of gravimetric observations in Shuvalovo Lake in winter 1927 and
1928,
Z. Geophys. Jahrgang V, Heft 7. p. 261285 (1929;).
34. Oltay, Karl. Die Genauigkeit der mit der Eotvos'schen Drehwage durch
gefuhrten relativen Schwerkraftsmessungen. Geodatische Arbeiten d.
Baron R. von Eotvosschen geophysischen Forschungen No. III, Budapest,
1928.