# SCIENTIFIC - EARTH MEASUREMENTS

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Document Number (FOIA) /ESDN (CREST):

CIA-RDP80-00809A000600231031-1

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RIPPUB

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C

Document Page Count:

19

Document Creation Date:

December 22, 2016

Document Release Date:

July 18, 2011

Sequence Number:

1031

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Publication Date:

July 5, 1949

Content Type:

REPORT

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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-
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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.
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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
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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.
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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
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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
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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
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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
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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
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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'
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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'
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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)
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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
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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
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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-