SCIENTIFIC - EARTH MEASUREMENTS

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CIA-RDP80-00809A000600231031-1
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RIPPUB
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C
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19
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December 22, 2016
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July 18, 2011
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1031
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July 5, 1949
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REPORT
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Sanitized Copy Approved for Release 2011/07/18: CIA-RDP80-00809A000600231031-1 INFORMATION FROM SUBJECT Scientific - Earth measurements HOW PUBLISHED Bimonthly periodical WHERE PUBLISHED Moscow (_- 7112101020.1' c011T01201NFO021T00 uncTin t-MWM~MiVrMso or 700 Wing sums 207010 1.2 is 110e Or 1VLa" Act 20 2.1 1.. 01 .10 30.00 "2200., in 7020/130100 00 7121171017100 .1 i71 000400711010130 "7 .013002 70 " 2300710/010 100"1 It 220?. 0101730 07 YB. 00011100 Or 71110 MAN is P20211310*. DATE DIST. Jr Jul 1949 NO. OF PAGES 19 SUPPLEMENT TO REPORT NO, Iavesti Akadeali Nauk 368,E Seri Goom-af i sb i Goof ieheska , n 3, 1 . Aerial Surveying and Cartography Moscow A study is made of the conditions necessary for the eolvalh;! bility and uniqueness of solution of an integral oquetlon by the aid of which the outer gravitational field and the shape of the physical surface of the earth may be determined, A. Listing's Geoid and the Quasi-Geoid An overwhelming majority of all geodetic surreys -- leveliing, trian- geoidal surface which is considerably more even. If these surveys esbracs surface wnica is acre convenient mnthe4eaticall.y. from a geoid Is inadequate. It has possible inherent con%sadiotions, since these discrepancies can noticecbly reduce the accuracy of results in soun- Sanitized Copy Approved for Release 2011/07/18: CIA-RDP80-00809A000600231031-1 50X1-HUM Sanitized Copy Approved for Release 2011/07/18: CIA-RDP80-00809A000600231031-1 5w16 In our opinion, the xc,:uction problem in geoleey is mainly complicated by the tradit+onal tendency to reduce observat'on7to one level, i.e.'to iitiryg s goo' ccincidirg with the average ocean level. Reduction to Bril10hinas geoid, enveloping the whole terrestial'mase but too far removed from a large part of the physical surface of the earth, is even more com- plicated. It must be admitted that reduction to Listings geoid Is a ra- tional procedure; but up to the present it bas been logical only so long as the results of gravimetric work were not applied in preparing geodetic observations spol triangulation by the method of "rectifying" geodetic lines from a geoid to a ellipsoid, while retaining their lengths; so that a rep- etition of the projection of geodetic systems fz,om a geoid to a ellipsoid would not hold true. But the possession of guvimetrie surveys, already made in considerable detail over large sreas of the whole earth, make ea- sentiel corrections possible in the methods of studying the shape of the earth. It Is also opportune to review the question of the expediency of re- taining the Listing geoid as a basic surface to whirrs or by means of which the geodetic elements measured are reduced to an ellepsoid. With reference to this question the following basic cirou stances must be considered: 1. The shape of Listing's geoid in generally not determinable, if density and disposition of the masses lying outside the geoid are not known. Therefore, the study of the sha teak Pe of Listing's geoid is partly a geological , since it cannot be solved, strictly speaking, before the completion of a geological Investigation Of all the continents. 2. Even with exhaustive geological data, a sufficiently accurate re- duction tc, the geoid Is connected with solving a complicated problem in the theory of potential because the reduction is carried out on an unknown sur- face of the earth, on which limited values are determined directly by ob- servation. 3. By means of triangulation, now used in the U$SR as a method of projrotion, the geoid is an intermediate reduction surface which must be excluded In the transition to a reference ellipsoid. Nevertheless, it leaves troublesome residual non-conformities because of the differences in reduction methods, 4. Geodeaiets are very rarely obliged to make observations of suffi- cient accuracy on the surface of the geese. The location for an observation post with rbepeet to a geoid is, etriotly speaking, not known because it Is only with a certain approximation that even orthcuetric heights can be con- sidered as measured from the surface of Listing's geoid. 5. It is not necessary to connect the basic scientific problem of higher geodesy with the study of Listing'a geoid. It would be moro desir- able to strive for the study of tl?e outer gravitational field and the ahape of the physical surface of the earth. 6. The shape of the physical surface of the earth can be determined with sufficient reliability on the basis rily of data obtained from exact measurements, ;.e., from the results of levelinge and of measurements of gravity related to points with well-known approximate actroncmic and geo- logical or geophysical data need, be involved in the principal solut on of this problem. This circumstance does not exclude their usefulness, for example, in the interpolation of gravity and on many other occasions. Sanitized Copy Approved for Release 2011/07/18: CIA-RDP80-00809A000600231031-1 Sanitized Copy Approved for Release 2011/07/18: CIA-RDP80-00809A000600231031-1 7. Knowledge of the shape of the physical aurfaca of the earth, under these conditions, supplies iivlispensabl.e data for the solution of all prac- tical problems of geodeey, particularly those originating in the high-pre- cision measurements of legrees. Problems In reducing all geodetical elements measured to an ellipsoid can, on principal, be correctly solved. The last two statements above will be corroborated later. However, the use of the geoid undoubtedly had one positive side: it divided the most inaccurate part from the unitary, extremely, " mplex, physical surface of the earth, a picture of which is given by certain .levelings, 1. e., height above sea level. There remained a second, inacmperably smoother part, I. e., height of the geoid over the ellipsoid. Such a division is very net- ural and rational, and.the geoid with both magnitudes divide4 hen an aM i- tional simple physical meaning. But it is known that when such a division is achieved in practice a great number of obscurities and insurmountable diffi- culties arise. Furthermore, it will be shown that a similar division of the torrential surface into irregular hypsometric and smooth geoidal parts can be made gradually, without recourse to Listings geoid, by examining a oer* thin surface close to the geoid. This surface, unlike the geoid, is deter- mined on the basis of employing only data from exact geodetic measurements without depenf:ing on this or: that notion o! the structure of the earth. As we shall see later, the surface in questi-a is characterized by a disturbing potential on the terrestiel surface, and its heights are ob- tained like the quotient by dividing the disturbing potential, at a given point of the earth4e surface, by the normal value of gravity, calculated in a corresponding manner for this point. For the sake of definiteness we are obliged to introduce a now term for the surface in question; let us agree to call it a quasi-geoid. In the problem under consideration the quasi- geoid is introduced to separate the lees smooth from the smooth parts of the earth. The former is determined by integration along the contour, and the second is obtained by solving a boundary problem in the theory of potential. On the ocean plane, the quasi-geoid coincides with i geoid but on continents the quasi-geoid can be taken, if necessary, as an approximate expression of the geoid ehepa. We mast consider, drat of all, how to separate the irregular part in the shape of the earth, which we shall call "the heig;it of ":ne point of the surface of the earth with reference to the quasi-geoid," or, more briefly, the "reference Z7 apomogatel'niy, literally anxiliarl/ height." It would be advisable to determine the reference heights so that they would be suffi- ciently close to the orthametric heights. However, the usual orthawtric correotlon does not entirely do away with the dependence of the result of leveling between two fixed points on the position of the guide line connect- ing 'them, which must have an effect on the dissimilarity in the heights when polygons of high-precision leveling are formed. The referonce heights eau easily . determined in a manner which will completely rid them of this defebt. B. Reference Heights Let us consider the normal potential field U, formed. by the "comparative earth" In which all masses are included inside the ellipsoidal am-face of the level, characterized by the dimensions of the semiazes a, a, b, the angular velocity of rotation w and the value of gravity on its equator Ys . The potential will nor be uniquely determined at any outer point of space by the coordinates of this point. It is 'cnrenient to select for our purpose the coordinates described below. Let us draw through the specified point a coordinated line which will be a line of intersection passing through this point of th, meridian plane and the hynerboloid focal to the leveled ellips- oid. The location of the point to be determined in space can then be described CONFIDENTIAL Sanitized Copy Approved for Release 2011/07/18: CIA-RDP80-00809A000600231031-1 Sanitized Copy Approved for Release 2011/07/18: CIA-RDP80-00809A000600231031-1 C4 ' 1AL by an angle, tazu3ort to this. line, (drawn at the Point of its intersection with the solid leveled ellipsoid) with the mane of the equator and the angle of the oerielien rlane, in vrh].c'!i this tango^:t lies, wi,h the piano of the original meridian, (latitude it*, longitude L*), and also length H* of the segment, of the line of force from the given point -co the leveled ellipsoid. We can regard the potential of the real earth W(87L-0, H-)-Wa-- ~ gd H as known (correct to an additive constant WO) at all points of the physical surface of the earth, and only on this surface; whereas it is not deter- minable at all other points of space without knowledge of the shape of the earth, and the density of the attracting passes, if it is a question of internal points. Thus, we can also foranalate an analytical expression for the disturb- ing potential T . W - U, but only for points of the physical surface of the earth. But since the shape of the earth is not known, the true coordinates of these points Be, L*, H* are unknown to um. However, we can consider the. approximate value of the coord.inates B, L, H as known, whereby the magni- tud ea _L-L (1) are so small that their second powers and products may be disregarded. For this reason, in all further calculations, only terms of the first order relative to d B, A L, C are retained. With these assumptions, let us expand the expression for the normal potential into the Taylor series: T(B*I ek H")= W113 L* Hk)-U(a*H" )= W,- f gd H-U (B, H)- u(~H) . au 13 H dB (2) aH a,3 In this expression we can disregard the term dUa1, H) A S t1 11") He C3, ayes because ev-n when d B = C;5 nn error less than 10-6 H will be introdu-ed. It is advisable to determine the subsidiary height 11 with the help of the equation - ~gdH=U(,g H) - U(Q, 0)=U(13,H)-Uo which has a simple (i) physical meaning. It indicates, namely, Chet in calculating the reference, height H by the difference of the potentials, we assume that the potential field of the earth is normal. Vndar conditon (3) we obtain from (2) a relation analogous to Brownie rill-known formula; T(C3e, L? /f*)- - a e B M) 4'? W~ - Uo ' y ($H)~`'-I-W where y(B,B) in the normal value of the acceleration of gravity at the point B, H, and the constant term Wo - IIo can, if desired, be reduced to zero by a proper selection of a surface of reference. Bgvation (3), as far as its left-ha* texr can be considered as known, can be used in calculating H directly, if an exact analytical term is employed for the normal potential rasalting fioai the theory of "Piteetti" and "Samil'yan'Z_tranelltarationJ. However, in order to facilitate calcula- tions, it is better to expand it Into a series which convergem quickly in this case; besides, there are sufficiently detailed tables for these coeffi- aiente. Sanitized Copy Approved for Release 2011/07/18: CIA-RDP80-00809A000600231031-1 , Sanitized Copy Approved for Release 2011/07/18: CIA-RDP80-00809A000600231031-1 an assIM with above-Monti o:,ed accuracy than: - Sy Jar- - Thug to deter^ci.ne B ?u+ us obtain the equation cy.~IH HaH-t z i4 y3ay~`~ Granting. when E x 10 ken, a relative erg or of not more then 1.10-6, in the terms of the second and third order, we need not consider cimpreaeion d Y(Q 0)=?'_ (1+9 sin~l3-Qo ssn'22 a/1 and R is the average radius of the earth. Starting from (5) it is not difficult to find a convenient corrective formula for o*.ar? calculations for converting the difference of the observed heights of two points into the difference of their reference heights: ~;,, I vv- . ,11 1 where, as we know: 1a4H+I (-r )4H+2(r;-r)H,4 l1a)J (6) or R +z (Y - Y2), uSI For the solution of equation (34), it is necessary that any solution of equation (15) should be orthogonal to the free term of equation (14), satisfying the condition: 1.[ Ysec a dS7 dS=0. (16) In fact, mautipplying equation (14) by,u and integrating over the whole surface B, we obtain: ',L tr f4-t4 ~r_seco_ [J r TY 'v I cos ?I d2.H-5 'x v,H)oosacJdS~~($=~NI Yseaad53dS. Changing the order of integration in the left-hand mile, which is possible because one of the integrals ld an ordinary one, end integrating the first with respect to the second argument, we find that if Ito satisfies equation (15), the left side will be reduced to zero and, consequently, condition (lo) should hold true. It is for this reason that the necessiti for this condition arises. Proving thb adequacy of condition (l6; in solving equation (14) ire much more ccemplicated. CW nIAL Sanitized Copy Approved for Release 2011/07/18: CIA-RDP80-00809A000600231031-1 Sanitized Copy Approved for Release 2011/07/18: CIA-RDP80-00809A000600231031-1 V. D. 1Gz adze's article on "Scene Peculiar Equations of Mathematical rhyaicak[3Juoted the bass, results of G. Gireud'e research. Girard proh&i for equations of x ci^-iler tyjc the fol.lotti;g. theorems, which are well-known from the theory of regular equations: . (1) A finite nuciber of linearly ir;Sependent solutions of a homologous equation corresponds to any pole of a resolvent; (2) A combined equation hoe the same number of linearly independent solutions; (3) The necessary and sufficient condition for solving a homologous equation is the orthogonal character'of the right-hand aide of the equation with respect to all solutions of a homologous combined equation. It follow from the last theorem that fulfilment of condition (16) is both necessary aM sufficient. Changing the order of integration in (16), ve can put it in the follow- ing form: fA(g-Y)sec oc dS=O, aA ~ I dSe ac cos cc A-car t ~4 Y ' 7y- + aH) D(4 H), U% Let us try to cmitjt from this equation since in condition (17) only X enters. For this purpose, lot us bear In mirul that Jl may be interpreted as the potential on the surface S of a simple layer with denaity/K, as shown by equation (18). Derivatives of the potential f a simple layer with respect to the direction of the tangents to the surface are continuous. The normal derivative of the potential of it sinpln 3,ayer on surface S experienoes a discontinuity; whereupon the discontinuity and the value of 6A on the surface are doter--inei by the well-known formula of Poisson and ?wimeil rtraaeliteration]: aa_a aorl as a s 2 ah= .0M. _ d?-, 31 aence the left-hand side of equation (15) equals The value of X a: 17 or. surface 8 are cowueectai with Creen's formula: -;-I_ ~ I', .-a-;L -- x dumskabMir d5, anJ Sanitized Copy Approved for Release 2011/07/18: CIA-RDP80-00809A000600231031-1 Sanitized Copy Approved for Release 2011/07/18: CIA-RDP80-00809A000600231031-1 Suubstitutirl; the right aide of (19) for- hero, we obtain a linear integrr..oI ,quati)n Which must be satisfied by the function of ;11r), d / 'Sr.c rx i- G'r O H - ea`sa ~fJl,ND d$? L 7r~~ It is Bay to simplify this equation. In fact through (12): s-- cu .ocda,HdS=?--5j H,Go. eR.dS. It follows from the defiiiition of the operator T) that: f5(-L, H)=A5(p,H)? 1 50,H)? Taking this into consideration and substituting for obtained according to (11), .+e chrll find that: a+ d)Vrte ?~ seeocc~,s? -Y acv Is. a linear integral equation with respect to (again with a special -A"' integral). We are now more Interested in the function of ? . In fact, for each aoluticn of (15) there is a single value of the fuucUius of A ," ' since asoiguaont of the surface simply determines the potential of a simple layer. On the other hard, search for t according to assigned values of A -chat Is, the density of the euriace leq'er according to the value of its po- tential on the surface amounts to the solution of Jirichlet'a outer and inner problem, since the density of the layer can be expressed by the differ- once between the outer and inner derivatives ant the potential of this layer. As we know, this problem is always capable of solution ard, more- over, of a unique solution when the hypotheses about the properties of sur- face S are cufficiently trued. fhus, k and eK are simply interconnected and the number of linearly independent solutions (15) act (20) are identical. The latter circumstance has an essential significance for us. Inasmuch as the number of linearly independent solutions of a homologous equation corres- ponding to th.? complete equation (14) and the number of such solutions of equation (15) combined with it are identical, it can be affirmed that the number of linearly independent aclutinne of homologous equation (20) Is equal to the number of linearly independent volutions' of the homologous equation obtained from l14) by eliminating the free term. Since the question of solutions for a corresponding homologous equation has an essential signi- ficance in studying the conditions for solving (14), we shall explain the number of ouch linearly independent solutions of (20). For this purpose, ?ot us set up an equation combined with (2cv). Proceeding analogously, as was done r-rlier, and calling the unknown function of the combined equrtion f , we shall have: 2'h ?r= sec. O? d J'-See eC? ~v ~26(21) Oonsidori?g ,) as the sirfacodensity of a simple layer :nt introducing the function , the potential of this layer at any point of .:urfece S. we obtain: ?a 27J Sec of , - '76C. CC curet Sanitized Copy Approved for Release 2011/07/18: CIA-RDP80-00809A000600231031-1 Sanitized Copy Approved for Release 2011/07/18: CIA-RDP80-00809A000600231031-1 potential of the simple layer with raspcc "rt. to the c orsaing.the :ii_ octioz P -~ 7 7X Ja r n.t? . id.1- ipaomuch as cos oe 0, inatea,i of (22) we shall obtain: -C -Y ~Y (24) This equation is equivalent to (21) in the sense that for each solu- tion of one of these equations there is a corresponding unique solution of another equation which, as before, follows from the uniqueness of the solu- tion'of Direchlot's probico. The solution Ir s 0 does not interest us. harmonic functions there is a corresponding solution, differing from zero, for the solution of the homologous equation obtained from (14). It is easy to see that this condition is sufficient 3ieectly from the original equation (14). Once the harmonic function satisfies (25), it also satisfies (24). This function will also satisfy (14) if the anomalies are correspondingly expressed. g-7~-av + r ' av T, (10) we may feel sure that when T = w , on th, basis of (24), g - Y = 0 and consequently ##is the volution of a homologous equation corresponding to the left site of (14). has the rireateet practical value, where the derivative is considered constant (the coefficient of reduction in free air), InV pure form it correepoa..is to the hypothesis that the surface of reference is a sphere. Thu,a, if the surface of reference is a sphere, Y4, where p is the to the macs of the earth. Since the potential w -)utside 2 should satisfy Sanitized Copy Approved for Release 2011/07/18: CIA-RDP80-00809A000600231031-1 Sanitized Copy Approved for Release 2011/07/18: CIA-RDP80-00809A000600231031-1 vhere11 is any spherical function of the first oilier. This means that far yv ve have three linearly, independent eblutionn, ai;ice an nrbitrary spherical function of the first uruer is the sum of three linearly independent func- tions. Hence, in this case equation (24) and, consequently, (21), (20), (10), and, finally, the homologous equation obtained from (14) have three linearly independent solutions apiece. E? Physical Meaning of Solvability of Conditions Thus, when the surface of reference is a sphere, there are three linearly independent fumei4ona 1(, ) and A, , which satisfy equation (20). Assigning to the surface the combined values g-Yfor the solvability of (14) should satisfy three conditions of the t;pe (17). The valuehas, in fact, been obtained from observations, measurements of gravity, leveling, and astronomical work. Will the conditions of (17) be fulfilled if the measure- ments are made with perfect accuracy? Ta answer this question let us carry the study of (20) further. Turn- ing to spherical coordinates lot no introduce instead of .. a new function . , connected A by the relation: xseeec=per (27) We shall have as a result: IF .1 'V No`, equation (20) can be pct in the form: ` 2IlvCosc:= cO. +P aEf JQS. / (28) Denting bye, the value of the rallus vector for a specified point and by V the angle betweento and (o , let us make use of the well-known relationship between these magnitudes ant e : r2'pa+P6 - 2PP0 Co Dr _PA. (P - e. aP - r3 dP- rc r 1 +Pa Pe in place of (28) we obtain: L fE J.C a Co- -'IO ' an Po 'r a)-r dj Pa Ac at-0 Sanitized Copy Approved for Release 2011/07/18: CIA-RDP80-00809A000600231031-1 Sanitized Copy Approved for Release 2011/07/18: CIA-RDP80-00809A000600231031-1 The tern 271ecoe cc on the left-hand aide equals half the gap ;.n the dieoontinuity of the derivative of the potential of a simple layer of density e- with respect to the direction forming the angle OC with the normal, that is, the direction Pe . Therefore, introducing the auxiliary function Ui, the potential of a simple layer of lensitycr. at a point lying outside 8, let us put (30) in the form. The outer derivative is taken off the left-hand side of the equation. The function Ui. =YiP (32) outside the surface satisfies Laplaws solution and equation (31). TLS T of Laplace, Y1, depends on three linearly independent functions, to each of which there corresponds a unique and completely definite value of the density s,- In fact, I T, r where Us is the solution of Direohlot's outer problem corresponding to the boundary value of U1iSince a single combination of values iicorresponds to the assigned combination of values for Ui on surface 8, A quo and fully definite value o.' or will correspond to each value of U1. It is possible to reach this result, if we study tic equation obtained from (30) and (32): s , c. c.s cf = Y,' -- .a . Sf d S, As a result of the transition from 'A to 0- , with the aid us pat the condition for solvability in the following fora: S,.(9-Y) Pd-5- o, or, eliminating, by the aid of (33), in the form: (3k) of (97) let raus_aua 1 a>, a7~ Y) pets. 0. Let us study the function V= P . (p'T). 'On surface Sand, by virtue of (10) V.- AV -0. (38) (39' Sanitized Copy Approved for Release 2011/07/18: CIA-RDP80-00809A000600231031-1 Sanitized Copy Approved for Release 2011/07/18: CIA-RDP80-00809A000600231031-1 C8 s If we assume the existence of three derivatives of the function T, that is, that the Volume density of the naseee generating potential T is a differentiable continua function of the coordinates, it can be shown that for the volume of a finite surface S S eY AVd-_ 0. Nov, making use of the equation PA (, "r) =,00 A T) IF readily verified by the aid of Iaplacee veil-known expression for the operator in spherical cooi9linatea, we obtain AV= 4A T+(0 C)P AT. (41) Substituting (41) in (40) and considering that: ralT ` padp ?1e+5 where dearrs is an element vi a ai 1ici angle, we f Ind that; eY,1QYd't ~y Y, I.TdeIPaI.~J~F4r~~p 4TJ do. 1stJJus convert the secmd integral on the right by integration by parts: e SaY ?Tdp; o c we shall then have: SpY'&Vdr=1MY1 ATdW. :jn the baei6 of tale hr ~thoel.e .)f continuity of AT, we must assume that d T.. 0 also or surface 8 and, therefore, that the right aide of the last equation `is been shown to be equal to zero; this proves the correct- ness of the assertion made in formula (40). let its study condition (36), writing it with the aid of (38) in the following manner: The functions V and Us, harmonic outside 8, are regular at infinity, and therefore: ~(Vo~_u~ fl Ls=a, MA1ng the l.eet and bMy....t-- e --~ two -q=t4- ,_a a:n. up aia,Q `w`ant aii~~s Ke. on the surface B, we eliainate the unknown funoticn and obtain: ah ~(V - Uiah )d5-o' 50X1-HUM Sanitized Copy Approved for Release 2011/07/18: CIA-RDP80-00809A000600231031-1 Sanitized Copy Approved for Release 2011/07/18: CIA-RDP80-00809A000600231031-1 sUg4/d r G. Converting this surface integral by means of Green's formula to a volume integral ant notiru the harmonic properties of, the funwtion.Ui But we have shown that Ui = Ylp,and therefore cnnclition (k2) coin- cides with the demonstrated equation (40). Just ee three arbitrary parameters form part of Y11, so all three conditions as we have seen, superimposed on the anomolies g - Y, are themselves satisfied. Thus, if the boundary values y--Y are not arbitrarily assigned and have a definite physical significance, being obtained by observations and, consequently, corresponding to some distribution of masses, the oonditione for the existence of a solution of our problem must be satis- fied; these conditions can govern the accuracy of observations from which the boundary values are obtained. The hypothesis of the differentiability of volume density can possibly be eliminated by further extensive demonstrations. But in this physical problem this hypothesis seems entirely permissible to us. In fact, the number of surfaces of discontinuity in the density inside the earth is finite end., consequently, in a sufficiently fine layer, it is possible to redistribute the masses in such a way that the density at any point can be differentiated. At the same time, the change of the outer potential field., as a result of the redistribution of masses, will be less than any previously assigned magnitude if a sufficiently small thickness of the layer in which the redistribution of masses takes place, is selected. the sphere of reference in now direction; it is evident that for this dis- plaoesent we are providing for three degrees of freedom. Nov the potential developed by the earth of comparison at any point of space in S and oatside 8 changes to the ma?vitude.L , where Y1 is Laplace 'a Y of the first order and dependent upon three oinetents: three coordinates of the center of the displace surface of reference relative to the original position of Its center. A now, value of the disturbing potential and a new value forEvill correspond to the new position of the surface of reference. of (14) will be equal to the sum: T- T,tTv The reason for the second term is obvious. In fact, let us displace Let T1 be the particular solution of (1k) and T2 be the general solu B. Conditions for a Unique Solution Returning to (1k), we can now verify the fact that when the boundary values are correctly obtained this equation can always be solved but that But thereby the sum uT + I ?-Cyr-Y) Sanitized Copy Approved for Release 2011/07/18: CIA-RDP80-00809A000600231031-1 Sanitized Copy Approved for Release 2011/07/18: CIA-RDP80-00809A000600231031-1 is not changed. Consequently, for the new position of the surface of reference: I A(TI e cads-t- ~(~)?~..)ds. After calculating therefrom equation (14), which by ag7sawant Is satis- fied by the function Tl, we may feel certain that the function y, satisfies the corresponding homologous equation. Therefore, Ts.~ ~a P'- aod the general solution of (14) is such that: ea. (43) The correctness of the result obtained is easily rerif?.ed by direct calculation, but we shall not do this Eliminating the condition U. = Wo in the last equation giving the general solution of the initial equation (14), ire obtain the general expression for C : (44) where Tl is the particular solution of (14) and Z1 is a derivative, spherical function of the first order. The physical sigai*ioanoe of the multiFliolty of solutions obtained is clear: here, dust as in the general vase, the dimensions ad position cf,.the surface of reference remain indeteisinate. Mnsequently, in this case also it is possible to introduce a coalition that the vol:mss of the quasi-geoid and the surface If reference be equal. Such a condition is geometrically more obvious than the equality of potentials Uo and Wo or the equality of the masses. Moreover, it in possible to demand the aos!!i- nation of mass centers or, main in the interest of geometrical obviousness, to oos-b.ne the centers of the volumes of tha quasi-geoid aid the surface of ref,.-once. It is easy to espresb these geometrical conditions in eaclyt- ical form. The oondition of volume equality ~ d~rsc1 'ICr Y permits determining Uo - Wo (tee teas with Z1 drops out). Here do, is an v1sownt of the sphere,* surface. The :orAltion forrrr combining th volume centers Tfi - t c cos p i )f Oc c0) y r a Sanitized Copy Approved for Release 2011/07/18: CIA-RDP80-00809A000600231031-1 Sanitized Copy Approved for Release 2011/07/18: CIA-RDP80-00809A000600231031-1 CON IDEN1 CCf M)MMIAL where 4t is calculated from ar. arbitrary point on.the sphere, permits definint* all three coefficients entering into n1. The magnitude of the second integral on the right Lido of the equation is negligible,, and hence the position of the center of the volume,can be dptermined almost irada- pendently of U0 ifo. However, when the volume centers are combined, the rotation axes of the reel earth and the earth of comparison do not coincide and the potential of centrifugal force ent*s, to a slight degree, into the disturbing potential. Hence it is better to demand the coincidence o1 the center of inertia of the earth with the center of the sphere of reference. This condition is expressed in the following f ~ATe j d T= 4, where Zl is any spherical function of the first order. By means of Oreen's formula the volume integral on the left-hand can be converted into a surface integral: raE JLa `?Z, 771- (e z,)] d 5'?' If we substitute here T from (43) and inclmxie as before sad utilize the fact that the three coefficlente entering into Zi are arbitrary, we obtain three equations for the determination of the three arbitrary constants enter- ing into (43)- G. Rena eaenting Earth, a Sh pe pr Density of the Surface Layer. Let us turn to equation (21) and examine it and the heterogeneous equation 29D-9pcesa-dy ~~dS-f' r dv S~dS69-Y- (45) It is evident that (20) will combine with it; but the condition of solvability, formula (17), is general for this equation and for the original equation (13). consequently all conclusions reached in studying (13) are fully applica- ble to (45). Let as introduce the auxiliary ilunction T' -- the potential of a simple layer located on surface 8 of :lenity qi The value on surface S derived from T' in the direction wr. forming the anglo (n, m) with the direction n of the outer normal to B, Is expressed by the well-known formula aMTi. S . 2.,7r f es(y1,m)? a7ft (47) BUSDOMix ADVICEN IIAL Sanitized Copy Approved for Release 2011/07/18: CIA-RDP80-00809A000600231031-1 Sanitized Copy Approved for Release 2011/07/18: CIA-RDP80-00809A000600231031-1 form: With the aid of (46) arsl: (47), equation (45) can be mitten in the a >,~ y d v r- (q - Y). Calculating (48) from (10) and following the reason ng given at the cad of Section D, we may feel certain that, in a case like that attdied.by use of a spherical surface of reference: (OA Comparing this result with (43), we reach the conclusion that any ?oL tion of (45), after substitution in (46), leads to the particular solution of equation (13). (13). fierce the two equations (45) and (46) are equivalent to equation For the two surfaces of reference we obtain by the aid of (29): ~Po +1P.._ ps, (49) ? tee r Now equation (45) is greatly simplified: s,_ 1 a r~cesa-