Below is the uncorrected machine-read text of this chapter, intended to provide our own search engines and external engines with highly rich, chapter-representative searchable text of each book. Because it is UNCORRECTED material, please consider the following text as a useful but insufficient proxy for the authoritative book pages.
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
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 every-day 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
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.
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 o-t the bar and therefore in the same horizontal plane. Figure 3 represents the block diagrams of the space of a Coulomb balance, or o-t all Mottos torsion balance; AEF and COG represent, respectively, the vertical planes o-t maximun~
~]P7~F ~S ~ r0~\ ~ 171 and minimum ping ABC represents ~ borizont~Ipl~ne~ KLI-II TepTeseDtatbeVertic~lcyliniertb~oughtbep~thoTs~ingoItbevei~t W;~,c,],representtbetr~cesoflevelsud~cesin Deplanes AFf aniCB08~]intbecylinierELl-II;~-TI-g-I-breprosentstbetIRceof 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
172 FIGURE OF THE EARTH spection of the blocl; diagrams, it is of course self-evident, 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. / </ a: / -_ ICON_, ~ an' \ :'N \: \ \ //, l FIG. 4. Block diagram of the space around an Eotvos balance representing the divergence of the level surfaces. That portion of the horizontal component of gravity which is a function of the differential warping of the level surfaces can be shown to be a function f(,a`J of the a%imut,h a of the point and of gravity, times the
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 saddle-shaped, 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, h-i-lt 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. .,
~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.
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~--g-sin~ cos a (4) bW=T sir1 ( 180°-a~ =-g ~ cos a si~i a 12 F=(rnl)(bV~:-aTV)=(rttl)(~( ~ - ~ Jisin2a (5)
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 28-to 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=~ and-R 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 U7-1' 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 US-UXX 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.
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 d-a-a' or d-e-a'' 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.
1','S FI GURE OF THE EAR TH b ,, / ~. . ~ ,/ a/ ,/ \~ 1 1 e c As' '\ d \\ FIG. 7.
GRAVITY MEASURE l1TENTS WITH TORSION BALANCE 179 acldin`~,, Therefore lVg+~7V'Gq (g) =Ng+l~g cos a. ~7Gr(9)-g COS a. (9 cos a) is a small horizontal force which acts on either weight of the ha.lance and which is set up clue to the i'act that above or below the level of' the center of' gravity, the vertical is not perpendicular to the horizontal anr1 gravity there-f'ore has the small hori7<onta.1 component (q cos a). This small horizontal force acting on one of the weights of' a torsion balance will term to rotate the balance. As the situation. in orbed' ;~ I. .svmmetrically reversed from that in o acda, the force acting on the upper weight of' an Eotvcis torsion balance is equal to, and opposite in abso- lute direction, but the same in rotational eff'ec~t as that acting on the lower weight. As g cos a = :Y,q? ~ q ~ ~ the torsion balance is able to measure the gravity gradient. The actual magnitude ~ of' the torque acting, on the weights of' the balance in any particular case is proportional to the sine of' the angular cliderence of' the azimuths of' the gradient and of' the balance. If' the azimuth oI:' the balance referred to the direction of magnum gradient as the axis of' reference is US, F=-ml1' Gr ~ q ~ sin A, where 7b iS the vertical distance between the wei~ht.s. (10) If' the axis of' reference is transferred to north, and if' in the clerivation of the transl'orn~ation equations the ',rarlient is consi~lered to be i.n tilde northeast quadrant, and if' a=the flew azimuth of the balance, and if y=the azimuth of' the gradient: h=.a-y, -1 F.=-ml1~Gq (q) (sing cosy-cos a siny). The quantities Go (~) cosy atoll Gq (g) siny are constants for any particular ~,rac1;e~t encl. represent the north-south and the east-west com- ponents of' the `,,~adie~.~t, respect;~Tel.~. In the notation of:' the calculus of' the ',ravitational theor\-, Go (g) cos Y = a a atoll Gr (g) sin Y = a a-, or in the abbre~:iater.l notation in use in torsion balance poorly Ups, and Up, respectively. Equation (:11) then felonies =-rn7~1 Us- s; n a + Phil UP COS a. (~12)
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 2a-1H 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 single-balance type of' instrument and then at least -five determinations *Actually a complicated mutual relation exists.
GRAVITY MEASUREMENTS WI TH TORSION BALANCE 181 of ~ are necessary. As the snorers double-balance 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 of-each 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 trt-roof = K Us sin 2a+KUx.~ cOs 2a-MUxz 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 table-lil~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 <gradient and of R and of their respective azimuths are then obtained by equations (13) and (8), which rather commonly are set in tabular or graphic form so that the calculator has merely to enter the table with the values of Us- and U-',. BUNG and Up,) and then to read off: the value of the gradient and the azimuth of the gradient (R arid A). The observed values at any station reflect the effect of topography and irregularities of mass close to the instrument. In field work with the Eotvos torsion balance, levels are run out to 20, 50, or 100 meters from the instru- ment, the distance depending on the irregularity and relief of the topog- raphy, and the effects on the gradient and differential curvature are cal- culated by simple tabular forms. In rugged country the ejects of the more distant topography have to be calculated with the help of topographic maps. The e-f3:ects of hidden irregularities of mass in the soil and subsoil
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<nowlecl~,e of' the actual value of' gravity. The form of' the equi.potentia1 (level) surfaces is determined by the . . a U a u au a u . ~ u at ~ ~X' By, az! ~ away A: those quantities, and- r epresen.t respect-ivel,v the reciprocals of' tile radii of curvature of the equipotential suri'aGe ill the x.z anti yz planes; a ~ represents the rate of' the vertical variation of p;ra~ity; and a U iS a factor arising from
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' astronomic-geoclet,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.
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), <Ada <fat, ¢~> a <au: 3.x Aft t3ya8XfaaU)' (4)aa (aaU)' (6)aa Bait, (8) as Bait, Ma ~aaUg, and the unit of' measuren~e~:~t for each of those quantities, in the c¢,s. system, is expressible in dynes per gram per centimeter. The anantiLv. MU a, which represents the rate of the variation o:t' the ~,ravit.ational poten , · ~· , ~, . ~ . .. . ,,, teal in the vertical direction is also the acceleration due to gravity and the quantities (1) and (2) represent respectively the rate of the va.ri- ation of' ~ravit.y in the ~ and the y directions, i. e., the components of' the gradient measured by the torsion balance. Quantity (8) is identical with (1), and (9) with (2~. The quantity (7) is the vertical gradient of gravity, with which the torsion balance is not concerned. The quantities (4) and (~) are identical and are, in a different ~na.thematical notation, the Ugly of torsion balance work. The difference between the quantities (3) and (6) is measured, but not the quantities themselves, and is the Urn of' torsion balance world. Empirically, the values of' Use, Use, Up, U$y observed in work with the torsion balance ra.n~,e most commonly from ~ to DOxiO-9 dynes per gram per centimeter with a mean maximum value of 15O x 10-9 Jvno~ rear .ornm nor o~ntim~t~r An +h~ All;+ 1 v in-g --~-~~~ A-- I--- rid ~ em ^~ an dynes per '~ram per centi~net.er, was the one used by Baron Eotvos and is the one in common use in torsion balance world today, Schweydar * has suggested that it be called an " Eotvos,'' abbreviated to " E." Although the significance of the R values lies in their indication of the cliffere~:~ce in the curvature of' the level surfaces in the two principal ' W. Schweydar. Z. Geophysics. Ja.hr=. I, Heft 3: 81 (1924).
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 five-tenths of one per cent, as the value of the reciprocal is about 1.02 x 10-3, 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 straight-line 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
~ ~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 irregula-ri- ties clue to salt comes, volcanic necks, ancl such moclerate-sized 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<l(liel:lt and curvature values therei:'ore represent the sum o-l' the effects of' the irregularities of the first five orders. The normal northward gradient and clift'erential curvature in Practice are cleter~i~ed .£ron~ tables which have been made up for tl~a.t purpose ally are subtracted from the respec- ti;-e observed values. If' areal and regional surveys are nude, the anomalies in the variation of' gravity of the t,ra.vity graclient, and of the curvature value consequent upon the irregularities of the various orclers, can he segregated by eliminating in order either the successively larger or smaller anomalies and then Cain be stucliecl. The .~:orn~al~orthwarcl, or l-~lanet.ar~, ,raclient which is the first-order eject of:' the greater value o-£ ¢,::avit~- at. the poles than at the equator,
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 north-south 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 (10-0 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 torsion-balance 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~eiss-schist-Ordovician limestone ridges o:t' North Texas, the Texas
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~sion-balance 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: 337-395. . 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: 639-682 (1923). 7. Shaw, H., and Lancaster-Jones, E. The Eotvos torsion balance. Phys. Soc. London, Proc. vol 35, pt. 3: 151-165 (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: 18-26, 86-93. 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. Lancaster-Jones 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.
GRAVITY MEASUREMENTS WITH TORSION BALANCE 180 13. Shaw, H., and Lancaster-Jones 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. Lancaster-Jones, 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: 39-74 (1929) . 20. George, P. W. Experiments with Eotvos torsion balance in the Tri-State 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: 763-778 (1929) . 22. . Control and adjustment of surveys with magnetometer or the torsion balance. Bull., Am. Assoc. Petrol. Geol., vol. 13, no. 9: 1163-1186 (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. Lancaster-Jones, 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: 149-154 (Aug. 1929~). 28. Heiland, C. A. Geophysical methods of prospecting. Principles and recent successes. Colorado School of Mines Quarterly, vol. 24, no. 1: 17-47 (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: 1-28. 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. 1-14. H. M. Stationery Office. 1928.
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: 1-32 (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. 261-285 (1929;). 34. Oltay, Karl. Die Genauigkeit der mit der Eotvos'schen Drehwage durch- gefuhrten relativen Schwerkraftsmessungen. Geodatische Arbeiten d. Baron R. von Eotvos-schen geophysischen Forschungen No. III, Budapest, 1928.
