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FË7A@-4/>L %°x*G7B-@-E+F2-C ™M0!J¡:„B-(±ŸA01­,6B-02B-(2C
>R%46(+J¡0uB0)4/7,/7A@%B-(2C¸$¯0!B->L P7A,/46,6.+02B->L'CÏ´P(+L@%4/7D0!B->LSB-¼H+('LL02J¡('*7AªB%>L…B0!B->L
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024Ë7A,6J†§'7,/H102L'"
 7A('H'(+,6QH+02:
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£@%7@-A02LP:0!A0L@0„H+2*G7,6*G@02E+F2-"
" ) J¡0uB-,6LQH+@-LQLAF2¼J¡02,/LSB%('7D0!46U02B-0vB02L…024/7,/7@-B-(+LTH'>J (+LQ:(',/7¼0v@%J]0L@-:(+?GF6H+,/(¡B%(¡Q(\N
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¦>0­,/B02B-(!"%>}('LQ7D002.1F!-CG(+LQ7(ŸH102: F/7A@%46=02:-('L('*G7A0@-J (+LQ@-J†H+>*%H+,6LQB-('LQ7A('LH+>*%H+(+, N
7A2L+"
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J¡02LL02L+C(uH+2J ˆ,6*­>('L’B-’¶G@02B-02B-ˆB0ˆB-,6L?7DÍ2*-H',<0¼('*G7((+4/(+L+C2@ LQ(#˜0%Cj0É£>QE10ÉB-(
0!7A02E+F2„¦>A01­,/7A02H+,/>*024‹('*7A(SB->,/LH+>Q:2L•§S:-Q>:2H+,/>*024‹02„:-B-@%7¡B-(SL@-02L)J¡02LQLA02LP(
,6*­2(+L02J†(+*7A(¯:%>:>H',6>*-02402¶G@02B%A02B-…B0…B-,/LQ7DÍ!*-H+,60S¶G@-(¯2L}L(':020 sQo! H C *IKI Q "
&:%46,6H+02*-B-ª’H'>*-H'(+,/7ˆ02H+,/J]0v:02A0¼B%@02L:02?7 F/H+@-4602L+C«>@¨:2*G7>LB-(ZJ¡02LL02L m ( m C
L(':0202B->LP:>@-J¡0B-,6L?7DÍ2*-H',<0 l C0,6*7(+*-LQ,6B02B%(SB-0£2E10B%(=0l7AA0!E1F2„¦>0­,Ë7D02H',6>*0!4 F §
( )
( )
1
F =K
m1 m2
,
l2
2
+“" *
L ('*-B- F 0u,/*7A(+*%L,6B-02B-(„B0Z£>E+0¶G@-(¡02¦>(†0246>*%¦>ZB0u46,6*%U0(+*7AQ(]>LTB-2,6LŸ:>*7A>LSB-(
J¡02LL02L m ( m ( K " 0H'>*-LQ7A02*7A(S@-*-,Ë­>(+QLA024Ϧ2A01­,/7D0!H+,62*024`"
šJR>A0v02L…J¡02LL02L m ( m LQ(Z0!7A02,<0!J B-(¡£2J¡0LQ,6J†§'7AQ,6H+0%Cϧ]H'>*­>(+*%,6(+*7(ZHDU02J¡02
@-J¡0¼B-(ZJ¡02LL0 i2sYQi$#qi (u0É>@%7A0¼B-(ZJ¡02LQLA0 i2swYQi2s_rmli "j¯@‹C—0!,6*-B0“CŠH+2*-L,/B-(+02*-B-!N L(u0
J¡02LL0†0l7AAL0 F6B0,/¦>@0240†@-J¡0@-*-,/B02B-(T(S0J]0!LLA0„0!7A0!7,/­!0,6¦2@024‹0 m >R%7A§'J„N L( 1
2
( )
1
2
F =K
79::EJ Y5
B9LA
%'& Ll>:A>A'587( 58<'NS1642
58= − 1727 1571 − 1630
)
K = 66, 7 × 10−9 cm3 g −1 seg −2 (SI)
m
l2
+“",+
G¶ @-(Q(+:-Q(+L('*7D0É0ª0l7AA0!E1F2v¦>A01­,/7A02H+,/>*024( %('H',6B0ª:(+460vJ¡02LL0 m LQ>R-(0v@-*-,/B02B-(¡B-(
J¡02LL0„B%,6LQ7A02*7A( l pH @[ (\q YH o 4 C *I M Q "
°x*7AQ“B-@%.+,6*%B-Ê@%J L,/LQ7A('J]0¨B-(ˆH'“>QB-(+*-02B02Lu(\7D02*%¦>@-4<0!(+Lu(!CP02,6*-B-0%CMH+2*-L,/B-(+02*-B-Ê02L
H+>QB-(+*0!B02LSB0vJ¡02LL0ª0!7A0!7,/­!0 m H+>J†L(+*%B- (ξ, η, ζ) (]02L=H'“2B-('*02B02LSB-v:>*7
0!7AN0 F/B- P (x, y, z) £Î¦>@%A0 +“" * C«0£>E+0ª:B-(†L('(':-('L(+*7A02B0É:2=@-J ­>(\7A> C«B-(
J¡02¦>*-,Ë7A@-B-( F (TH+>J†:2*-(+*7A('L (X, Y, Z) B02B02L:>'C
x−ξ
Km x − ξ
×
= −Km 3 ,
2
l
l
l
Km y − η
y−η
Y = −F cosβ = − 2 ×
= −Km 3 ,
l
l
l
z−ζ
Km z − ζ
= −Km 3 ,
Z = −F cosγ = − 2 ×
l
l
l
X = −F cosα = −
L('*-B--C
l=
p
(x − ξ)2 + (y − η)2 + (z − ζ)2 .
žŠ,6¦2@-A0 +“" * ™}2J¡:>*-('*G7(+L•B-(
°x*7AQ“B-@%.+,6*%B-„¡('LH10!4<02
W
F
££2E10B-(S0!7A02E+F2 pH
@[
(\q Y
H
o 4
+“"/.
+“"1-
C *I M Q "
B-('Î*%,6B-„:> W (x, y, z) = K
m
,
l
+“" D
¼¶G@0245(':-('L(+*7A0’ |olsxX\e^\_i ‚ ‘2Yi2m_rs`i^\_`olei ‚ >@¨B%( i2swYQi o=Xsxole_i2ei >¦ (+02B-É:>
@-J¡0¡:0!Q7 F/H+@-460¡B-(…J]02LQLA0 (ξ, η, ζ) C*G@-J :2*G7 (x, y, z) CB-,6L?7D02*7A( l B%( C 02L&H'>J„N
:>*-(+*7(+L
(X, Y, Z)
:02LQLA02J 0„L('PB02B02LP:>
X=
™}>J *%27D02E+F2­>(\7A>Q,<024wC
∂W
;
∂x
Y =
@[
YH
(\q Y
H
∂W
;
∂y
o 4
Z=
C *I M Q ∂W
.
∂z
+“"/M
+“" Q
= (X, Y, Z) = gradW ;
>@uL(x˜0%C„­>('7>£>E+0§S ‘2YQiq2_`X\esxX B0£@-*-E1F!†('LH+024<02 W "
šLQ7A0¯H10202H'7(+P F/LQ7,6H10•B-¯('LH10!4<02˜Q@-LQ7,/ÎH+0¯0)LQ@0¯@%7A,/46,/.102E+F2)*%¯46@-¦>02¸B-¯­>(\7A> "!«2—>@%7
4<0!B--CH+>*-LQ,6B-('A02*%B-@-J L,6L?7A('J]0…B%(¯­2¤!,62L5:>*7A>LJ¡0!7(+,602,6L m , m , · · ·, m C“B-,6L?7D02*7(+L
('L:(+H\7A,/­!02J†(+*7( l , l , · · ·, l CG=:27(+*-H',<024-727D024B%…L,/LQ7(+J¡0S§)0=LQ>J]0ŸB0!L}H+2*G7,/R-@-,6E'O“('L
,6*%B-,/­,6B%@02,6L YH @[ (\q YH o 4 C *I M Q 1
1
2
2
n
n
W =K
n
X
mi
i=1
li
+“"/0
.
“› @-:>*-B%¶G@-(ª02L„:02Q7PF6H'@-4<02L„('LQ7A(x˜02J B-,/LQ7AQ,6R-@F6B02LH'>*7A,/*@-02J¡('*7A((+J @-J ­2>46@-J†( v C
H+2JKB-('*-L,/B02B-( ρ = dm/dv C(+JK¶@%( dv §Ÿ@%JK('46(+J†(+*7B-(&­>246@-J†(¯( dm @-JK(+46('J¡('*7A
B-(TJ¡02LL0%C-†L>J¡0!7AÀ2,6…L(Ÿ7A02*-LQ£2J¡0(+J @-J¡0,6*7A('¦>A0!4 W =K
ZZZ
v
dm
=K
l
ZZZ
ρ
dv,
l
+“" I
+“" *C
v
L('*-B- l 0B-,6L?7DÍ2*%H+,<0('*G7(T„(+4/(+J†(+*7A†B-(TJ]0!LLA0 dm = ρdv (S„:>*7A¡0!7A0NF6B% P "
š “:-46,/H+,/7A02*-B-„LQ,6LQ7(+J¡0B-(TH+>B-('*02B0!B02LPH102?7A('L,<0!*02L>R%7§+JNxLQ(
W =K
ZZZ
v
ρ(ξ, η, ζ)
p
dξdηdζ
(x − ξ)2 + (y − η)2 + (z − ζ)2
L('*-B-†(+4/(+J†(+*7A†B-(S­>>4/@-J¡(TB0!B-]:> dv = dξdηdζ C“]¶G@-(S(“:-4/,6H10†0¡,/*7A(+¦2A024‹7,6:%4<0%"
 :!7A(+*%H+,<0!4 W §&H'>*7 F/*@%(+J 7AB-('L:0!E+(&7(+*-B-(¯0 klX\YAo *%,6*%Î-*-,/7 YH @[ (\q YH o 4 C *I M Q "
&LŠB-(+Q,/­!02B02L:-Q,6J†(+,602LB-( W Cl,6LQ7&§!C>02L«H+>J†:2*-(+*7A('L—B0)£2E10“Cl7A02JR§+J LAF2&H+>*7 F/*G@02L
(+J 7“B-&&(+LQ:02E'-C2:>§'JvCl02LŠB-(+Q,/­!02B02LL('¦>@-*-B-02L*F2)L(MH'>J¡:>?7D02J B0)J¡('LJ¡0•£>QJ]0%"
½)2L—:2*G7>L—>*%B-(0&B-('*-L,/B02B-(5­20!,<0•B-(+LQH+>*7,6*G@02J†(+*7A(P0!L—B-(+Q,/­!02B0!L—L('¦>@-*-B0!LŠ7D02J…R§'J
02:-Q(+LQ(+*7D02J B-(+LQH+>*7,6*G@-,6B-02B-(+L'"‹°xLQ7Z“H'>Q(:>¶G@-(]:27(+*-H',<024 W LA0!7,6LQ0!.0(+¶G@0!E1F2
B-(S«2,6LLQ>*
( )
*-†,6*7(+,/>MB0!L•J¡02LQLA02L YH @[ (\q
YH o 4
C *I M Q ∆W = −4πKρ
(+J¾¶@%(
∂2W
∂2W
∂2W
+
+
.
∂x2
∂y 2
∂z 2
∆W =
³LQ,6JR>4/
∆
+“" *K*
+“" * +
+“" * .
+“" * -
HDU02J¡0lN L(T>:(+A0!B->4<0!:-4<02H',<02*%…(Ÿ7A('J 0£>J¡0 ∂2
∂2
∂2
+
+
.
∂x2 ∂y 2 ∂z 2
ž>A0B%¡H'>:†0!7A0!7A,Ë­>-C-*%†('L:0!E+„­20!.+,6%C0B-(+*-LQ,6B0!B-( ρ §T.+('-C4/>¦> ∆W = 0.
š LQ7A0u§„0Z('¶@-02E1F2uB-(„·¸02:%4<02H'( (LQ@02LTL>4/@-E+O(+LŸLQ(HDU02J¡02J £@%*-E+O(+LŸU0!J†¢>*-,6H+02L¯('L?N
£§+Q,6H+02LLÀ>4/,6B0!L+"%)LLQ,6JC-B-,6.tNxLQ(S¶G@-(T„:27A('*-H+,6024‹¦>0­,Ë7D02H',6>*0!4§T@-J¡0…£@-*-E1F!¡U0!J†¢>*-,6H+0
£>0„B-02LPJ]02LQLA02LP0l7AA0l7A,/­!02L'"
( )
)
B
B
£>QE10]¶G@-(„0l7A@0]LQ>R-Q(@-J H'>:Z(+J (':2@-LZ*0LQ@-:(+QGF/H+,/(=7A('Q(+LQ7(§0Q(+LQ@-4/7A02*7A(
B0…£>QE10¦>01­“,Ë7D02H',6>*-024(ŸB0…£>E+0H+('*G7PFË£@-¦0B-('H+>Q(+*7(SB-027A02E1F2…B0©«('A0“"% £>QE10
7A!7D024wC>@ˆLQ(#˜0%CÏ0ZQ(+L@%4/7D0!*G7(B02LŸB-@-02LT§HDU02J¡02B0 olY iªqX=‘2YQi2m_qiqX " |olsxX\e^\_i ‚ qX
‘2YQi2m_qiqX >@ ‘XDoD|olsxX\e^\_i ‚ C W g C§…0]LQ>J¡0†B-2L&:27A('*-H+,602,6LP¦>0­,Ë7D02H',6>*0!4`C W C(…H'(+*7A ËF N
£@-¦>%C W C
+“" *D
Wg = W + W
L('*-B-†¡:27(+*-H',<024‹H'(+*7A F/£@-¦2†B%('Î*-,/B-„:2 c
c
1
Wc = ω 2 (x2 + y 2 )
2
[ ROh*T8LEAf;5VA'RU7gP EL RU7979LEA PjRO5V=K=95Va [ ROhLEAs :E'
< NO:JM5 1781 − 1840
1749 − 1827
+“" * M
(+J¿¶G@-( ω §0&­>('46H+,6B-02B-(02*%¦>@-4<0!Š( px + y ¯0 B-,6L?7DÍ2*-H',<0&B%Ÿ:>*7T02Ÿ(+, %ŸB%(27A02E1F2%"
›“@-R%LQ7A,Ë7A@-,/*-B-„*0 +" *ED 02LP(+¶G@02E'O“('L +“" I ( +“" * M >R“7A§+JNxLQ(
ZZZ
+“" * Q
dm 1
Wg = K
+ ω (x + y ).
2
2
2
l
2
2
2
v
$&(',/­!02*%B-„†:27(+*-H',<024‹H'(+*7A F/£@-¦2†('J ('4<02E+F2„0
∆Wc ≡
(x, y, z)
C
+“" * 0
∂ 2 Wc ∂ 2 Wc ∂ 2 Wc
+
+
= 2ω 2 ,
∂x2
∂y 2
∂z 2
(H'>JR-,/*02*-B-]H+>J 0(+¶G@02E+F2ZB-(>,6LQL>* “+ " *N* :0!A0
B-(S«2,6LLQ>*:02A0„:27A('*-H+,6024‹B-(T¦>0­,/B02B-( W C
W
C2R%7A§'J„N L(0('¶G@02E1F!Z¦>(+024
g
+“" *I
∆Wg = −4πkρ + 2ω 2 .
&:%46,6H+02*-B-„>:(+02B->¦2A02B-,/(+*7A(S02„:27A('*-H+,6024
= gradWg ≡
Wg
∂Wg ∂Wg ∂Wg
,
,
∂x ∂y
∂z
+“" + C
,
> R%7§+JNxLQ(& m2X\sxolYŸqXP‘2Yi2m_qiqX ¶G@-(¯Q(+:-Q(+LQ(+*7D0„0=£>QE10=727D0!4
r£>E+0…¦>0­,Ë7D02H',6>*0!4J¡02,/L
£>QE10]H'(+*7A F/£@-¦0 ¶G@-(„0!7@0¡LQ>R-Q(0]@-*-,/B02B-(=B-(J¡02LL0 YH @[ (\q YH o 4 C *I M Q C
H+@˜02LPH+>J†:2*-(+*7A('LPL(S(“:-('LL02J :2 ∂Wg
gx =
= −K
∂x
ZZZ
x−ξ
ρdv + ω 2 x,
l3
ZZv Z
x−η
ρdv + ω 2 y,
3
l
v
ZZZ
∂Wg
x−ζ
gz =
= −K
ρdv.
∂z
l3
∂Wg
= −K
gy =
∂y
v
+“" + *
¡J 02¦>*%,/7A@%B-(PB-( g CGHDU0!J]02B-0…B-(&¦2A01­,6B02B%(2CG7A(+J 0…B%,6J†(+*-LF2=GF/L,6H+0SB%(¯@-J¡0…02H'(+46('A02E+F2
(„§¡J†(+B%,6B0Z(+J gals "« ­!024/>¯*G@-J†§+Q,6H+B-( g §¡B-(†02:% “,6J¡02B02J†(+*7A( I QN0 gals *%
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( )
e
e
1 2 00
1
(r R + 2rR0 ) =
R
J
∂J
1 ∂2J
∂2J
+
cotθ
+
∂θ 2
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+“" + D
,
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B-( θ ( λ VG>LdB->,6Lj4602B->LLQ(+AF!…,6¦2@02,6Lj0T@-J¡0SJ†(+LQJ]0TH+2*-LQ7A02*7A(2"&LQL,/JvCG0T(+¶G@02E+F2 +“",+ D :“B-(ŸLQ(+•L(':02A0!B0(+J B-@0!L•2R%7A('*-B-!N L( +" +NM
r 2 R00 + 2rR0 − n(n + 1)R = 0,
L('*-B-
n(n + 1)
∂2J
1 ∂2J
∂J
+
+
cotθ
+ n(n + 1)J = 0,
∂θ 2
∂θ
sen2 θ ∂λ2
*I M Q 0 q H (
YH @[ (\q YH o 4 0H'>*-L?7D02*7A(
"
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s T8LEA
=KRONSNOLE_'ROA
1889 − 1969
C
+“",+KQ
( Y
q (
C
[ (
R1 (r) = r n
+“" +L0
R2 (r) = r −(n+1) .
+“" + I
Š 0!A0]B-(\7A('J†,6*02)02L)L>4/@-E+O(+L):02A0¡0](+¶G@0!E1F2 +“" +NQ 0!:-46,/H10lN L(2C*%­!02J†(+*7(2C¡J¡§\7AB-
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(sen2 θ/gh)
+",. C
C%>R%7§+JNxLQ(
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+“"/. +
h00 (λ) + m2 h(λ) = 0.
+“"/.K.
h1 (λ) = cos(mλ)
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m2
g(θ) = 0;
senθg (θ) + cosθg + n(n + 1)senθ −
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00
0
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c
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s
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[ (
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H
∞ X
n
X
o 4
C *I M Q +“"/. Q
c
s
[Anm Ynm
(θ, λ) + Bmn Ynm
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∞
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n=0
n
X
c
s
[Anm Ynm
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W
§
+“"/.K0
m=0
∞
n
X
1 X
c
s
W (r, θ, λ) =
[Anm Ynm
(θ, λ) + Bmn Ynm
(θ, λ)] ,
n+1
r
n=0
m=0
e
+“"/. I
¶G@-(LAF!¯L>4/@-E+O(+LŠ:02A0Ÿ0)(+¶G@02E+F2TB-(·¸02:-4602H+(!C!Q(+L:(+H\7A,Ë­20!J¡('*G7(2C>:-02A0¯0!L—('¦>,6O(+L«,6*7A('*0
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nm
W (r, θ, λ) =
∞ n
X
a n+1 X
n=0
r
m=0
0
mn
0
[A0nm cos(mλ) + Bmn
sen(mλ)] Pnm (cosθ).
+“"1- C
š LL02L)LQ>46@%E+O(+L7AI'J @-J¡0(+:%(+LQ(+*7D0!E1F2¡¦>(+>J†§'7,/H10¶G@-(T§T£(',/7A0B-,/­,6B%,6*-B-„0('LQ£(+0¡('J
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+"/- C >R%7A§'J„N L(
"
#
∞ X
n
X
a n
KM
1−
W (r, θ, λ) =
[Jnm cos(mλ) + Knm sen(mλ)] Pmn (cosθ) ,
r
r
n=0
m=0
+“"/- *
L('*-B- a
A0
KM nm
a
0
=−
Bnm
.
KM
Jnm = −
Knm
+“"/- +
&½ 0„:-¤!7A,/H10§=J†(+4/U->*G@-J†(+Q,6H10!J¡('*G7(T@%7A,/46,6.+02M>LPH'“(\ÎH+,/(+*7A('L&*->QJ]0!46,6.+02B->L}¶@%(…02LQL,/J
L(TQ(+4<0!H+,62*02J (
J¯nm
1
J¯n0 = √
Jn0 ,
2n + 1
(
)
) s
Jnm
(n + m)!
(m 6= 0)
=
2(2n + 1)(n − m)!
Knm
K̄nm
+“"1- .
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i2_6{ >@ {\~|X\Y #`^\_`XD{¡qX†e #rm2X ‚ C‹¶G@-(†L(†H10!A02H\7A(+Q,6.+02J :>S02:-('L('*G7A02…uJ†(+LQJ¡:!7A(+*%H+,<0!4
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B-('*->J†,6*02B-02LB-(T¦>(+2:('L . H %qYHIs C *INIKI C
W (x, y, z) = W0 = const.
+“"1-K-
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H10!J¡:B-(T¦>A01­,6B-02B-(Ÿ7A('('LQ7( ¦>('>:(T£@%*-B02J†(+*7D0!4 @[ pH C *I Q D "
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dW =
+“"1- D
∂W
∂W
∂W
dx +
dy +
dz,
∂x
∂y
∂z
H+2J *-27A02E1F2­2('7A2,<0!4`C%@%LA02*-B%¡„:-Q“B-@“7A†(+LH+024<0!+C%7(+JNxL(
dW = gradW · dx = g · dx,
™}>*-LQ,6B-('A02*%B-TT­2('7>
'& m
58A'58D'ROJMBgs RO7@B9ROA'k
sendo
dx = (dx, dy, dz).
+“"1- M
02Ÿ4/>*-¦>ŸB-0¯L@-:(+?GF6H+,/((+¶G@-,/:27(+*-H',<024 (W = W ) C>T:!7A(+*%H+,<0!4
dx
& u & a & & t
0
§TH+>*%LQ7D0!*G7(T(
dW = 0
C-02LLQ,6J
+“"1-Q
g · dx = 0.
žŠ,6¦2@-A0 +“" D ™}@%Q­!0!7A@%A0„B-(TL@-:('GF/H+,6('LB-(T*
F/­2(+4
“› (S¡:%B-@%7A†(+LQH1024602PB-(SB->,6L­2('7A2(+L)§T.+(+Q-C>LP­>('7>('L&LF2¡:(+Q:('*-B-,/H+@-4602('L•('*7A(=L,w"
¹(+¶G@0!E1F2 +"/-Q (“:-Q(+LL0=H'4<0202J¡('*7A(M¶G@-(P¯­>(\7A>jB-(¦2A01­,6B02B%(M§*->QJ]024%»ŸL@-:(+?GF6H+,/(
(+¶G@-,/:!7A(+*%H+,<0!4‹¶G@-(T:02LL0¡*-„J†(+LQJ¡„:>*7A-"
024Ë7A,/7@-B-( H B-(P@-J :>*7S§J†(+B%,6B0Ÿ02T4/>*-¦>ŸB0Ÿ4/,6*-U0¯B%(•:%@-J† H'@-Q­!0 :02?7A,/*-B-ŸB-
¦>('À>,6B-( rΦ>@-0 +“/" - R "›“(T„­>(\7A> dx ('LQ7A,Ë­>(')0!†4/>*-¦>B-0†4/,6*-U0=B-(T:-@-J†-C-*-0†B-,/('E1F2
H+Q(+LQH+(+*7(…B-( H C-„H'>J¡:%,6J†(+*7L(+¤ +“"1- 0
| dx |= dH,
H+2J B-,/(+E+F2ˆ>:>LQ7A0É02É­>('7>„B-(Z¦>0­,/B02B-(!C g C—(Z’Í2*%¦>@-46É('*7A(
02:-4/,6H+02*-B-„0B-(\Î*-,6E+F2„B-(T:-B-@%7¡('LH+024<02M7(+JNxLQ(2C
g · dx = gdHcos180◦ = −gdH,
L@%R-LQ7,/7A@%,6*-B-†0(+¶G@0!E1F2 +“"1- M *-0 “+ "/- I ÎH+0 dW = −gdH
ou
g=−
∂W
.
∂H
dx
(
g
§uB-(
180◦
+“"1- I
+“" DLC
š LQ7A0¡(+¶G@0!E1F2¡§S£@-*%B02J†(+*7D024Ï:-02A0¡0†B-(\7A('J†,6*02E+F2„B0]0!4/7A,Ë7A@-B%(2C-Q(+4<0!H+,62*02*-B-„>LPH'>*“N
H+(',/7>L¦>(+2J¡§\7A,/H+>L'C H C-(TB-,/*Í2J†,6H'>L+C W YH @[ (\q YH o 4 C *I M Q "
V
)
B
=  J†B-(+4/B-(+*-2J¡,/*02B- X\Y\YQi SolYt[†i ‚ §@-J (+46,/:-LÀ2,6B-(—B%(dQ('­2>46@-E+F2P¦>(+02B-P:(+4<0Q27D02E+F2
B-(’@-J¡0h('4 F/:-L(ˆ(+J 7>*%ÊB- (+, “ÊJ†(+*-2+C02 ¶G@024TL(0!7,6R%@-,Ÿ0hJ¡('LJ¡0 J¡02LL0 (’0
J†(+LJ¡0…­>('46H+,/B02B-(ŸB-(TQ27D02E+F2„B0©«('0†Q(1024w"% L@-:(+?GF6H',6(ŸB-„(+4/,6:-LQÀ>,6B%(&7A('J :!7A(+*%H+,<0!4
H+2*-LQ7A02*7A(2C U C(§=,6¦>@-02402:27A('*-H+,6024«B%(…¦>0­,/B02B-( W CL>R%(0]LQ@-:(+QGF/H+,/(…B-¦2(+À>,/B-(
$s 4 (Go! C *IKI M "
 :27A('*-H+,6024B-(¦>A01­,6B-02B-(B0’©(+QA0’*->J¡024wCŠ7D02J…R§'J HDU02J¡02B-’B-((+LQ£('>:27(+*-H',<024wC
H+2J¡:-Q(+('*-B-(T¡('LQ£('>:27A('*-H+,6024‹B-(S0!7A02E+F2„(T(+LQ£('>:27(+*-H',<024‹B%(SQ27D0!E1F2-C
0
0
+“" D *
Ug = U + U c ,
L('*-B- U ,6¦2@024‹0 W B-(\Î*-,/B-„*0†('¶G@02E1F! +“" * M "
K('LQ£(+Q>:27A('*-H+,6024jB-(]0l7AA0!E1F2-C U C¶G@-(¡@%J ('46('J¡('*G7vB%(¡J¡02LL0 dm(x , y , z ) :%­>H10
(+J¾@-J :>*7A P (x, y, z) LQ,/7@02B-†0@-J¡0„B%,6LQ7AÍ2*-H',<0 l §T(“:-Q(+LQL]:> c
c
0
U =K
ZZZ
0
0
+“" D +
dm
.
l
v
&$ (T02H'>B%H+>J YH @[ (\q YH o 4 C *I M Q :-¤2¦-". + ,/*­>(+QL„B0B-,/LQ7AÍ2*-H+,60%C (1/l) C
B-(\Î*-,6B%†('J¾7A('J†>LPB-(Ÿ£@-*-E'O“('L•U-02J†¢>*-,/H102LM.'>*02,/L§
+“" D .
∞
1 X r 0n
=
Pn (cosΨ).
l
r n+1
n=0
+( ¶G@02E+F2 +" D . /, *7AB-@-.†*0 +“" D + >L=:>4/,6*-¢2J¡,/>L&B-(]·Ï(+¦>('*-B-(¡(¡LQ@-:>*-B-¼¶G@-(]02L
:02?7 F/H+@-4602LM(+L?7A(x˜02J -B ,6L?7AQ,6R-@
F/B02LMH'>*7A,/*@-02J¡('*7A(TL>R%(T@-J¾­2>46@-J†(2C%>R%7A§'J„N L(
U=
K
r
ZZZ
v
K
dm + 2
r
 7(+QJ¡„B-(T¦>02@u.+(+Q-C
ZZZ
K
r P1 (Ψ)dm + 3
r
0
v
U0 =
K
r
ZZZ
dm =
ZZZ
r 0 P2 (Ψ)dm + · · ·.
+“" D -
+“" DND
v
KM
r
(':-('L('*G7A0ɪ:27A('*-H+,60245B%(0!7A02E+F2ª:-Q“B%@-.+,/B-ª:>…@%J]0(+LQ£('A0¼U-2J¡>¦2I+*-(+0vB-(¡J]02LQLA0
C*-…:>*7A P CB%,6LQ7A02*7A( r B-…LQ(+@H'(+*7AQB-(&¦2A01­,6B02B%(2V0=J¡02LL0=B0T7A('A0S*->QJ]024wC M C
M
§T,6¦>@-024‹0J¡02LL0†B07(+0†Q(1024G˜¤†0!B-,6H',6>*0!B0»J]02LQLA0B0†0!7AJ†>L?£(+A0“"
v
¯ L7A(+QJ¡2L†B%(v:-Q,6J†(+,/É(uL('¦>@-*-B-¦>A0!@-L¡Q(+:-Q(+LQ(+*7D02JCj('L:(+H'7,/­!02J†(+*7A(!CjJ¡>J†(+*7>L
(+L?7D¤!7,6H'>L(TB-(T,6*-§'H+,60 . H qpHIs C *IKIKI "
™}>*-LQ,6B-('A02*%B-vu7A('J†uB-(7A(+QH+(',6¦>02@‹C‹:0!A0u( “:-('LL¤lNx4/v('J H+>B%(+*02B-02L=('LQ£§',6H+02L
,6*7B-@-.\N L(Ÿ0!L}£@-*-E'O“('LU02J†¢>*-,/H102L5B-(ŸL@%:('QGF6H',6(¯¶G@-(ŸLF2H+2*-LQ7,/7A@ F6B02L}B-(¯U02QJ¡¢2*-,6H+02L
.+2*02,6L ££@-*%E1F2B%( θ (…B02L&£@-*%E+O(+LŸ02LQLH+,602B02L)B-(·Ï(+¦>('*-B-Q(2C‹('LQ7D0!L¯B-(+:(+*%B-(+*7A('LTB-( θ
( λ " L,6J†('7,60ªB-ˆJ¡B-('46ˆ02B-!7D02B-É,/J¡:%46,6H+0v*0É*G@-4/,6B02B%(ZB->LL('H'7>,602L„(7A(+LQL('A02,/L+C
:>,6L5*F2:“B-(&U01­>(+B%(+:(+*-B-I'*-H+,60B046>*%¦>,/7@-B-(2V(+L?7D0('LQ7,6E+F2LQ(Ÿ(+LQ7(+*-B%(T7A02JR§+JK»2L
£@-*-E'O“('L¯02LQLH+,602B02LPB-(…·¸('¦>(+*%B-(=B-(…¶G@024/¶G@-(+•¦>A0!@‹" .+>*0!4B%¡7A('H'(+,6Q¡¦2A02@vH+>*7A§'J
L2J¡('*G7(T:27I+*-H',<02L,/J¡:-02('LB-( θ V
+" D M
P30 = (5cos3 θ − 3cosθ)/2,
(T„L(+@vL,/*024‹B-(':('*-B-(SB04<0!7,/7@-B-(2"
™}>J†ˆ0¼©«('0¼*->J¡0245:>LQL@-,}L,6J†('7,60vH+2J ('4<02E+F2É02É:%4<02*-É('¶G@0!7A2,<0!4}0É(+¶G@0!E1F2
+" D - L(T('LQ7AQ,6*-¦2(T02>LP.+>*-02,6L:0!(+
L . H qYHIs C *IKINI U=
KM
r
1 − J2
a 2
r
P2 (θ) − J4
a 4
r
L('*-B-†L(+J†,ËN (+, %…J]02,/>MB0©(+QA0„*->J¡024w"
¯LPH+('ÎH',6(+*7(+L LF2¡H+0246H'@-4<0!B->L0!7A01­>§'L)B%(
P4 (θ) − J6
a 6
r
P6 (θ)...
2n
J2n = (−1)
n+1
5nJ2
1−n+ 2
e
3e2n
(2n + 1)(2n + 3)
+“" D Q
+“" D 0
L('*-B- 0’( “H+(+*7,6H',6B0!B-(vB%('46,6:%LÀ>,/B-(( = 10.826, 3 × 10 CB-(+*-2J¡,/*02B-É0!7>
B-,/*Í2J†,6H+…B-(Ÿ£>J¡0%"
«B-(tNxL(T('*G7AF2†(+('LH'('­2(+&0†('¶@-02E1F2 +" D Q B0L(+¦2@-,6*7A(Ÿ£>QJ]0 −7
2
U=
"
KM
r
1−
∞
X
n=0
J2n
a 2n
r
P2n (θ)
!#
+ 0, 5(ωrsenθ)2
+“" DLI
 H'“(\ÎH+,/(+*7A(02:0!(+H'(u*-¥›,6LQ7(+J¡0ÉB-(v™}>*-LQ7A02*7A('L¡)LQ7AQ>*-¢>J†,6H+02LB0ˆ°xP °x*7A('?N
* 0!7,6>*-024j&LQ7>*%>J¡,/H1024j •*-,62* (†02.]:0!Q7A(¡B0ªB-(\Î*-,/E1F2¼B%>LL,/LQ7(+J¡02L=¦2(+B-§+LQ,6H+2LB-(
(\£(+QI+*-H',<0%CjH+2J¡-C:>„( “('J¡:-4/-C±S´)›0 C `±Ÿ(+B-('7,6H´P('£(+Q(+*-H'(ɛºL?7A(+J *I 0 C . H^a
qpHIs C *IKIKI "
 ('46,6:%LÀ>,/B-( ;±=› 0L- >Q46B ±Ÿ(+B-('7,6H’›ºLQ7(+J *I 0N- CM02B-27A02B- :('46hL,/LQ7A('J]0 ±Td›C
7A('JÁ:02AÍ!J¡(\7A2L•,/B-I+*7,6H+2L&022L)B-Z±S´)›0 C CLQ(+*-B%¶G@-(… ¹±S›0N-¡@-LA0]@%JÁ­!024/>P:020†
02HDU0l7D02J†(+*7A=¶G@-(&>R“7A,6B%…0…:-02Q7,6jB-….'>*024U0!J†¢>*-,6H'=LQ(+¦>@%*-B-¦>02@ B0…LQ(+¦>@-,/*7A(
£>QJ]0 q C + CKCKC +“," M C
=−
(5)1/2
7D02R(+460 “+ " * J†>LQ7A0>LP:0!AÍ2J†('7>LB-;±=›0N- q C+ CKCKC "
©«02R(+460 +“" * ›“,6L?7A(+J¡0¦>('“B-§'L,/H+„J@-*-B%,<024 *I 0N-Nd©«('0†½•>J¡024
¿L
GÏ
¸\ . ›“('J¡, Nx(', “J¡02,6>
= 6378137, 0metros
™}>*%LQ7D0!*G7(=±T0­,Ë7D02H',6>*0!4
= 3986000, 9 × 10 m /s
>*-024ϲ&0!J†¢>*-,6H'B-( + ¦>A0!@
= −0, 484166774985 × 10
HDU0l7D02J†(+*7A
α = 1 : 298, 257223563
8
3
◦
B
−3
>¦ A01­,6B0!B-(P*->J¡024*0SL@-:(+?GF6H',6(PB-…('46,/:-LÀ>,/B-(B-('B-@-.+,/B0=:>5›“>J†,6¦246,<0!*0
(“:-Q(+LQLA0¡('J¾£@-*-E+F2†B04<0!7,/7A@%B-(Ÿ¦>(+B-§'L,6H+0 φ C-§
( )
('J *I + I C
+“",M *
aγE cos2 φ + bγP sen2 φ
γ0 (φ) = p
,
a2 cos2 φ + b2 sen2 φ
L('*-B- γ 0u¦>A01­,6B-02B-(*->QJ]0!4—*-(+¶G@02B-2+C γ 0u¦2A01­,6B02B%(†*%>J¡024Š*%v:À>46%C a ( b L('J¡,Ï(', “„J]0!,6>5(SJ†(+*%>B-„(+4/,6:-LQÀ>,6B-(!C-('L:(+H\7A,/­!02J†(+*7(2"
Š0!A0¥7A02*-LQ£2J¡02Z0¥(+¶G@0!E1F2 +“/" M * *0¥£>J¡0B-(™M02LQL,6*%, @-L02B0¨:020¨B-('Î*%,6Z0
:-Q,6J†(+,60¼£À>J@%4<0É,6*7A('*0!H+,62*024B0¦2A01­,6B02B%(v('J *I . C C (cos φ) £>,P( %:%(+LQLh:>
(!C-B-(+:>,/LMB-(T024/¦>@-J¡02L}>:(+02E+O(+L'C%7A§'J„N L( 0 q H ( ( Yq ( [ ( C *I 0KM (1 − sen φ)
E
P
(
)
2
2
L('*-B-
f
1 + (f¯ − f − f f¯)sen2 φ
γ0 (φ) = γE p
1 + (f 2 − 2f )sen2 φ
+“",MK+
+“",MN.
†02HDU-0!7D02J†(+*7†( f¯ ¡02HDU0!7A02J¡('*7A„¦>A01­,6J†§'7,/H+-CB-(\Î*-,6B%>LP:> a−b
a
γ
−
γ
P
E
.
f¯ =
γE
f=
$&('L(+*­2>4/­2(+*-B-ɏB-('*->J†,6*02B%>Ÿ(+J LQ§+,/(„B-(¡:27I+*-H',<02LT(¡02R02*-B%>*02*-B%v>LT7A('J†>LTB-(
$
:=KNSLc[ Lh*ROkENORU:A>: '&+# u a . RSL Z :EA'A'R fWm $:E7K7@ROA'R # t a "$ t >QB-(+J LQ@-:(+,/>+C“7A(+JNxLQ(
+“"/MN-
γ0 (φ) = γE (1 + αsen2 φ + βsen2 2φ)
L('*-B- α = f¯ ( β = f (f − f¯)/4 "
½)†,6*7@
F/7¡B-(S@-*-,ËÎH10!)0†B-('Î-*-,6E+F2¡B-0¡¦>0­,/B02B-(T*->QJ]0!4`C0†°x•± £°x*7A(+Q*0!7,6>*0!4Ï&LLQ!N
H+,60!7A,/>*’2•±Ÿ(+B-(+L?º 02B-27>@‹C(+J *I . C C@%J]0u£À>J@%4<0u¦>(+0245Q(+H'>J¡('*-B02*%B-¼¶G@-(]£>LQL(
@%7,646,/.102B-0…*->L7A02R024/U->LMB%(S¦2A01­,6J†('7AQ,<0%C
γ30 = 978, 0490(1 + 0, 0052884sen2φ − 0, 0000059sen2 φ)Gal.
+“"/M D
š LL0„£À>QJ@-460…ÎH+>@H+2*-U-(+H',6B0„H+2J¡„0 ‡ lYt[~ ‚ i\esxX\Ytei2sw_`olei ‚ qi Æ Yi2m_qiqXSqX C
(¡­,/¦>>2@0!7§ *I M Qu¶G@0!*-B-ª£>02J 02:-Q­20!B->L…2L…*-­>2L…:02Í2J†('7AQ>LS:02A0¼(+4/,6:-LQÀ>,6B%(
¦>('“H'I+*7AQ,6H+%"³¦>0­,/B02B-(T*->QJ]0!4Ï*-]('46,6:%LÀ>,/B-(T,6*7A('*02H',6>*-024Ï:02LQL>@ª0¡L(' 0 q H (
( Yq( [ ( C *I 0NM γ67 = 978, 031846(1 + 0, 005278895sen2φ − 0, 000023462sen4φ)Gal.
+“"/MKM
š LL0*-­20†(+¶G@02E+F2]£>,HDU02J¡02B0†B-( ‡ lY\[~ ‚ i
\esxX\Ytei2sw_`olei ‚ qi Æ YQi2m_qiqXqX C(
£>,‹@“7A,64/,6.+02B0…:0!A0„0B-('Î*%,6E1F!¡B-¡›“,/LQ7(+J¡0¡±Ÿ(+B-§'L,6H'„B-(S´•(\£(+I'*-H+,60B-( *I M Q“"
šJ *I 0 C 0T°x•±C(+J¬LQ@0…02LQL('JR-46§',<0Ÿ¦>('A024-('J ™M02JR(+QA0%C)@-LQ7A¤24/,<0%C!(+H'>J†(+*-B-2@¡@-J¡0
*-­!0£À>J@%4<0:020¦>A01­,6B0!B-(2C
γ80 = 978, 0327(1+0, 005279041sen2φ+0, 0000232718sen4φ+0.0000001262sen6φ)Gal.
(+¶G@02E+F2†:02A0„0¦>0­,/B02B-(Ÿ*-2J¡024‹('4<0!7,/­!002;±=›0N-¡§
L('*-B- 1 + ksen2 φ
γ = γE p
,
1 − e2 sen2 φ
k=
bγP
− 1.
aγE
q
C+ CKCKC +“",MKQ
+“",MN0
A7 02R('4<0 “+ ",+ 02:%(+LQ(+*7D002L¯H'>*-L?7D02*7A('LŸGF6L,/H102L)(¦>('>J¡§\7AQ,6H10!L&(+*­2>4/­,6B-02LŸ*0(+¶G@0!E1F2
+," MK0
©Š02R(+460 +“" + ™}>*-L?7D02*7A('L•GF/L,/H102L(T¦>('>J¡§\7AQ,6H10!LM:020„ ;±=› *I 0L- q C + CKCKC "
l ¸¸ vl }¸
G v¸ «w¸
›“(+J†,ËN (+, %J¡02,/>
6378137, 0000metros
›“(+J†,ËN (+, %J†(+*%>
=
3142metros
±T01­“,/B02B-(Ÿ*%¡('¶@-02B-> γ ==6356752,
9, 7803253359m/s
±T01­“,/B02B-(Ÿ*%¡:À>46
8321849378m/s
š %H'(+*7AQ,6H',6B02B%(
γ==6,9,69437999014
× 10
2
e
2
p
2
)
B
−3
)
¹:-Q,6*-H',6:024%02:-4/,6H10!E1F2¯B%ŸH102J†:ŸB%(•¦2A01­,6B02B%(*->J¡024§P*0ŸB-('7(+J†,6*-02E1F2&B- |ols#X\e^\_i ‚
i2
e l[†i ‚ o >@ |olsxX\e^\_i ‚ |X\Y\s~Y+ŽDiqolY C T Cl¶G@-(M§MB%('Î*-,/B-¯H'>J¡&0&B-,Ë£(+('*-E10¯('*7A(}>L—:27A('*“N
H+,602,6LB%(=¦>01­“,/B02B-(T:-Q“B%@-.+,/B->L•*@%J :>*7¡:(+460†©(+QA0¡Q(1024 (W ) (S:('4<0©«('0¡*->QJ]0!4
C(T7(+J 0LQ(+¦>@-,/*7A(Ÿ£>J¡0 0Qq H ( ( pq ( [ ( C *I 0KM . H qpHIs C *IKIKI (U )
g
g
+“"/M I
T (x, y, z) = Wg (x, y, z) − Ug (x, y, z).
« B-(vL(+¡H+>*%L,6B%(+A0!B-H'>J¡’:27(+*-H',<024P¦>('A02B-:('4<02L [†il{D{+il{vi2e
l[†i ‚ il{ C—­,6L F/­2(+,6L†(
,6*­,6L F/­2(+,6L'C}¶G@-(É7A02*-L?£>J¡02J 0h©(+QA0¨*->QJ]0!4&*0¨©(+0¨(10!4`CM0!JR02LZH'>J 0¨J†(+LJ¡0
J¡02LL0%C%­!02,602*-B-„0!:('*02L•0†LQ@0„B-,6L?7A,/R-@-,/E1F2 . H %qYHIs C *INIKI "
&*-0246,/LA02*-B%T02L5('¶G@02E+O(+Ld¶G@-(•(+:%(+LQ(+*7D0!J 2Ld:27(+*-H',<02,/L(10!4(•*%>J¡024-H'>*-LQ7A0!7D0NxL(P¶G@-(
P:!7A(+*%H+,<0!4H+(+*7P F/£@%¦>P§d•J†(+LJ†P:020•0P©(+0•Q(10242(5:0!A0P0•©(+QA0P*->QJ]024!(5B%(+LA0!:02('H+(
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( Yq( [ ( C *I 0KM "Š™}2J ,/LLQ-C«ª:27(+*-H',<024j:(+?7A@-QR02B->=:B-(¡L(+…B-(+LQ(+*­>>4Ë­,6B-É('J
L§',/(SB%(T£@%*-E+O(+LPU02QJ¡¢2*-,6H+02LM(+L?£§+Q,6H10!L+C02LQL@-J†,6*-B%¡0LQ(+¦>@-,/*7A(T£>J¡0 g
g
T (r, θ, λ) =
∞ X
a n+1
n=0
r
+“",Q C
Tn (θ, λ),
L('*-B- T (θ, λ) 0£@%*-E1F2„U0!J†¢>*-,6H+0B-(TL@-:(+?GF6H',6(ŸB-(T¦>02@ n "
›“>R%(T„¦>(+À>,/B-(2C%¶@%(=*0„02:% “,6J¡02E1F2('LQ£§',6H+0†H'>Q(+LQ:>*%B-(S»†('LQ£(+0
n
T = (r, θ, λ) =
∞
X
n=0
Tn (θ, λ).
r=a
C%7§+JNxL(
+“",Q *
)
H
B
 ­>('7>j02*-2J]024/,<0PB-(¦>0­,/B02B-(M§M0¯B-,/£('(+*%E10Ÿ(+*7AQ(¯­>(\7A>B-(¦2A01­,6B02B%(2C g C2*-T¦>('À>,6B-(!C
;
(T­>('7>PB-(T¦>A01­,6B-02B-(T*->QJ]024wC γ C*%†('46,/:-LÀ>,/B-(¯B-(TQ('£(+QI+*-H',<0 £Î¦2@-A0 +“"/M C
+“" QN+
∆ = g − γ Q.
>¦ 01­“,/B02B-(]:B-(LQ(+J†(+B-,/B0ª@%7,646,/.102*%B-v­!¤2Q,6>LS7A,6:>L=B-(Z,6*-L?7AQ@-J¡('*7A>LB%(J¡('B-,6E+F2
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(
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H q^4YH p
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)
C + CKC . ½&0:-¤!7A,/H10B-(\7A(+QJ¡,/*0lN L(Ÿ„(+LQH1024602+C0!*->J¡0246,60=B-(T¦>0­,/B02B-(!C¶@%(S§TB0!B-¡:> +“" QL.
∆g = g − γQ .
L ('*-B- g 0]J¡02¦>*%,/7A@%B-(=B%]­>(\7A>¯B%(¦>A01­,6B-02B-( g =| | *-Z¦>('À>,6B%(…( γ §0]J¡02¦>*%,/7A@%B-(
B-…­2('7>}B-()¦>A01­,6B-02B-(&*%>J¡024 γ =| γ | *0=L@-:(+?GF6H+,/()B%(+4/,6:-LQÀ>,6B-(!C*0…J†(+LQJ]0S4<0!7,/7A@%B-(
B0>R-LQ(+?­20!E1F2†B0„¦>A01­,6B0!B-( pq (pH HIq^4 pH p[04po
pH C + CKC . "
Q
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P
=
P
+“" Q -
− γP ,
>*-B%( §)=­>(\7A>}¦>A01­,6B0!B-()B-0=©(+0…*-:>*7A…«C( γ …­>(\7A>5B-(&¦>01­“,/B02B-(•*->QJ]0!4
*-„J†(+LJ†„:2*G7-"-©«02JR§+J :-02A0(+L?7A(SH102LQ-C:“B-(tNxLQ(ŸH+>*-LQ,6B-('A02P„(+LQH1024602 P
P
+“" Q D
δgP = gP − γP .
) ) B
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‘XDol_qi ‚ C N rΦ>@-0 +" Q "
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qi¡m2X\Yts_`^i ‚ H+2J¡:>LQ7:('4<0H+2J¡:>*-('*G7(TJ¡(',6B%,<02*0 ξ C-B-,/('E1F2„*->?7A(tNxL@%4 ξ = Φ − φ,
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L('*-B- η = (Λ − λ)cosφ.
+“",QNM
. H %
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C *INIKI +" QKQ
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UP = U Q +
T g γ
N
§TH+>*-L?7D02*7(=('*G7(T†(+46,/:-LÀ2,6B-(¯(Ÿ†¦>('À>,6B-(!C
U
∂U
∂n
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:“B-(ŸLQ(+•(+LQH+,Ë7D0
N=
Q
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N = UQ − γN.
T
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P
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(WP − UP )
T
.
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γ = gradU,
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g=−
∂W
,
∂n
γ=−
∂W
∂U
− 0
∂n
∂n
+“" Q I
C
[ (
+“",0 C
= gradW
δg = grad(W − U ) = gradT ≡
∂T ∂T ∂T
,
,
∂x ∂y ∂z
.
+“",0 *
+“",0K+
∂U
∂U
≡−
,
0
∂n
∂n
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(+¶G@0!E1F2 +“," Q D LQ(+¤†B0!B-„:> 0
δg = gP − γP = −
) Hj=KA>7 B p
5VROA'=KROJCl $=9_'A 7
&
v &a
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−
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L('¦>@-,/*G7(Ÿ£>J¡0 h
δg = −
∂T
.
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∂T
.
∂h
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Š0!A0„0¡0!*->J¡0246,60=B-(T¦>0­,/B02B-(¯7(+JNxL(
1 ∂γ
∂T
−
T + ∆g = 0.
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+“",0 D
+“"/0KM
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02:-Q %,/J]0!E1F2T(+L?£§+Q,6H10“C“:>j( “(+J†:-4/-CG02HDU0l7D02J†(+*7A=B0=©(+QA0…§•B-('LH+2*-L,/B-(+02B--C…¶G@-(
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∂T
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02*-2J]024/,<0SB-(T¦>A01­,6B0!B-(¯7A>Q*02JNxLQ( [0X ( HIs C *IKI Q T (R, θ, λ) =
∞
X
Tn (θ, λ)
n=0
+“",0N0
∞
1X
δg(R, θ, λ) =
(n + 1)Tn (θ, λ)
R n=0
∞
∆g(R, θ, λ) =
1X
(n − 1)Tn (θ, λ),
R n=0
(“:-Q(+LQLA02*-B%u02L¯02*%>J]0!46,<0!LMB-(…¦2A01­,6B02B%(=('JÁ7(+QJ¡>L•B-(=L§+Q,6('L&B-(T£@-*-E'O“('L&U02QJ¡¢2*-,6H+02L
B-(TL@%:('QGF6H',6( X
+“/" 0 I
∆g(θ, λ) =
∆g (θ, λ).
∞
n
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[0X ( HIs C *IKI Q n=0
∆gn =
+“" INC
n−1
Tn .
R
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T =
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Z Z
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σ
#
Z Z "X
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π n=0 n − 1
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ž«,6¦>@-0 .%"/. ™}>B-('*02B0!L•('46,/:-L>,/B-,6H+02L
q p
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"
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s 4 (Go! C *IKIKI [ YH q Q4 pH C *IKIKI "
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(2C}02,6*%B0%CjL(¼H'>*-U-('H+,/B-¥­!024/>¡B%(¼¦>A01­,6B-02B-( g *-(+LQL(ÉJ†(+LJ†4/“H+024`Cd0’B-,/£('(+*%E10¥B-(
:27A('*-H+,6024 δW :“B%(ŸL(+•H10!46H+@%4<02B0B-0„LQ(+¦>@%,6*7A(Ÿ£>QJ]0 £J]02,/LMB-(\7D024/U-(+LP*-„H10!
: F/7@-46 + .“" *
δW = −gδh.
>*-B%( g §Ÿ>R-LQ(+?­20!B--"
«2§+JCH+>*-LQ,6B-('A02*%B-:>*7A>L¯H'>*-('H'7D0l7A>L¯:>•@-J¡0]46,/*-U0„B-(*-,Ë­>(+4602J†(+*7A-C
.%," + :>( “(+J†:-4/-C0B-,/£('('*-E10„B-(T:!7A(+*%H+,<0!4‹(+*7AQ( ( §T( “:-Q(+LQLA0]:> A B
CAB = WB − WA = −
™}>*-LQ,6B-('A02*%B-¡,/*7A(+?­!0246>L}B-,6LQH+(\7A>L
4h
Z
AB
B
gδh.
£Î-¦>@-A0
.%",+
.%"/.
B-†*-,/­2(+4<0!J¡('*G7„0†('¶G@02E1F! .%",+…ÎH10 WB − W A = −
B
X
A
A
g4h.
°xLLQ]LQ,6¦>*-,ËÎH10„¶G@-(…†*-,Ë­>(+4602J†(+*7A†H+>J…R-,6*0!B-„H+>J J†(+B-,/B02LPB-(S¦>A01­,6B-02B-(S:->:>QH+,62*0
B-,Ë£(+('*-E10B-(¼:27A('*-H+,6024`Cd¶G@-(¼§¼@-J¡0¶G@02*7,6B02B%(ªGF6LQ,6H10“CdR-¤2L,/H10('J 7“B007A('>,60B-(
024Ë7A,/7@-B-('L YH @[ (\q YH o 4 C *I MKQ "« (+L?7D0ÉB-,/£('('*-E10¼B¤NxL(ZÉ*%>J¡(]B-(Z*O-J¡('
¦>('>:!7A(+*%H+,<0!4`C C "
 *O-J¡(')¦>(+>:27(+*-H',<024:B-(dL('—(+L?7D02R(+4/(+H+,/B-¯0):02Q7,6B-(}@%J¿:>*7A P LQ>R-(}¯¦>(+À2,6B-(!C
;
(TL('•( “:-('LL¡:2 0Qq H ( ( pq ( [ ( C *I 0KM YH @[ (\q YH o 4 C *I M Q 0
Ci = −(Wi − W0 ),
L@%R-LQ7,/7A@%,6*-B-†0(+¶G@0!E1F2 %. ",+ *0 “. "/- Î-H10 Ci =
Z
Pi
gdh =
P0
Z
Pi
g0 dh 0 .
.%"1-
.%" D
P00
& L…,6*7A('¦>A02,/L…('L:(+H+,ËÎH10!J >L:%H+(+B%,6J†(+*7A>L02¼462*-¦>¼B0v*->J¡024wC dh CŠ(02É46>*%¦>ªB0
46,/*-U0B-(T:-Q@-J†-C dh rΦ>@-06.“"/- 0 q H ( ( Yq( [ ( C *I 0NM "
³@-*-,6B-02B-(=02B-27A02B0¡:-02A0†¡* O-J†(+Q¦>(+2:27(+*-H',<024ϧ kgal × metro "³A0!.1F2¡:02A0†(+L?7D0
H+2*G­2(+*-E+F2„§¯¶G@-(¯LQ(+@­!024625*G@-J†§+Q,6H+…*-(+LQLA0@-*-,/B02B-()§¯:-A0l7A,6H+02J†(+*7A(),6¦>@0!4»0!4/7A,Ë7A@-B%(
B-(Ÿ@-J :>*7A†02H',6J¡0…B-„
* FË­>(+4ÏB%†J¡02'C%(+J¾J¡(\7A2L+V-J¡02,6L5:-('H+,/LA02J†(+*7A(!C C =. 0.98H "
H
šLLQ()­!02462d§•:>L,Ë7A,/­2=¶G@0!*-B-…=:2*G7=('LQ7,/­>('M02H',6J¡0ŸB-…¦>('À>,6B%(2C.+('=*-…¦>('À>,6B%(•()*-(+¦>0lN
7A,Ë­>u02R-02, “B-Z¦2(+À>,/B-(2V§H+>*-L?7D02*7(:02A0:>*7A2LŸL,/7@02B-2L¯(+J @-J]0¡J¡('LJ¡0L@-:(+?GF6H+,/(
(+¶G@-,/:!7A(+*%H+,<0!4 0Qq H ( ( Yq ( [ ( C *I 0KM "
½&0É:-¤!7A,/H10%C«*-(+J h *-(+J g LF2H'>*-U-('H+,/B->LH+2J¡ˆ@-J¡0ª£@-*-E+F2H'>*7 F/*@-0ˆB-0ˆ:>LQ,6E1F!-C
H+2*-L('¶ %(+*7A('J¡('*G7(2C02L,6*7(+¦>02,6LMB-0„('¶G@02E1F!
.%" D *F2„:“B-('J¾LQ(+P01­!0246,602B02L0!*0246,Ë7A,/H10lN
J†(+*7A(!CLQ(+*-B-]*-(+H'(+LL¤2,/Z('H+>Q(+&0¡@-J¡0¡B%,6LH'(\7A,6.+02E1F!]¶G@-(('J¡:-Q(+¦0„­!0246>Q(+L•J¡('B-,6B%>L
B-( g (&B-( δh 02…4/>*-¦>=B0…4/,6*-U0T*-,Ë­>(+4602B0 0 q H ( ( pq ( [ ( C *I 0KM qYH$[ ps C
*I M C C“>R%7A('*-B-!N L( .%/" M
. X
ḡ δh ,
C =
0
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k
L('*-B--C
i
k
k=i
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1
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2
=§ 0†B%,/£(+Q(+*-E+0¡B-(S*
FË­>(+4Ï2R-L('Q­!02B0†(+*7AQ(…>L•:2*G7>L•02Bl˜02H'(+*7A('L+C (P ) ( (P ) C( g
§T­20!46>MB-0†¦>01­“,/B02B-(ŸJ†(+B%,6B0*-†:>*7A P rΦ>@-0 .“"/- "
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:0207AB->L]>L:>*7A>L'VM0 O-*-,/H10( “,6¦>I'*-H+,60§¼¶G@-( g C5¶@-0246¶G@-('¶G@-(ÉL(x˜0¨0¥L@0£>*7(2C
7A('*-U0ªR0:-('H+,6LF2-"Š (+L:-02E102J†(+*7É,6B-(+024jH+>Q(+LQ:2*-B-(+*7(u»vB-,Ë£(+('*-E10vB-(]
* FË­>('4`C δh C
(u0¼:-Q(+H+,/LAF2ÉB%>L­!0246>Q(+LB%( g B%(+:(+*-B-('J B0É*0!7@-('.10ÉB-¼7(+Q(+*-ˆ(ZB0ª­!02Q,<02E+F2¼B-
H10!J¡:B-(T¦>A01­,6B-02B-( 0Qq H ( ( Yq ( [ ( C *I 0KM "
δhk
k−1
k
k
k
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q
H (
( Y
q (
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C *I 0NM "
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L('@¨@-LªB-,6Q('7Av*F2v§H+>*­2(+*-,/(+*7A(!VŠB-('LL(ZJ†B--C¸>L=­!024/>('L=:B-(+J L('…( %:%(+LQL>L„('J
@-*-,/B02B-('L)B%(…B-,/LQ7DÍ!*-H+,60%CB-,/­,/B-,6*-B%†¡* O-J¡('¡¦>(+>:27(+*-H',<024¸:>P@-J ­20!46>PB-(S¦>A01­,6B-02B-(
H+2*G­2(+*-H',6>*-024 f YH H [5VBp:N m C + CKC + "
Hi =
Ci
.
g0
.%"/0
20 4/7,/7A@%B-(>R%7,6B0T*0=('¶G@02E1F! .%"/0=B-(':('*-B-()B-=7,6:SB%(&¦>0­,/B02B-(P(+*­2>4/­,6B-0%"G&LLQ,6JCL(
H+>Q('L:>*-B-('50Ÿ@-J¿­20!46>—H'>*-L?7D02*7A(•B-(P¦>0­,/B02B-(*-2J¡024`C!:02A0Ÿ@-J¿:2*G7=¶G@0!46¶G@-(+'C
g
0’024/7,/7A@%B-(Z§uB-,/7A0 q2_re 2[_`^i "›“( g £>,/¦>@024»ˆ¦>0­,/B02B-(u(+024PJ¡§'B-,<0¼('*7A(v¦2(+À>,/B-(
(ZÉ:>*7AÉ>R-LQ(+?­20!B--C0É024Ë7A,Ë7A@-B-(¡B-('7(+QJ¡,/*02B0ª§u0 olYtsxol[]€\swYt_`^i V—(2C«Î*024/J¡('*G7(2C«L( g
£>=,6¦>@0!450v@-J ­20!46>SJ¡§'B-,6vB-(¦>01­“,/B02B-(¡*->QJ]024j('*G7(Z0v(+LQ7A02E1F!¼(Z0ªLQ@-:('QGF/H+,6(¡B-(
(\£(+QI+*-H',<0%C0„024Ë7A,/7@-B-(¯2R%7A,/B0§THDU02J¡02B0B-( eolYt[†i ‚ f gH H$[5MBY:N m C + CKC + "
&L•024Ë7A,Ë7A@-B-('LMB-,/*Í2J†,6H10!LMLAF2†>R%7,6B0!L:> 0
0
0
HiD =
Ci
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γcte
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(+¶G@-,/:!7A(+*%H+,<0!4B%…H10!J¡:=B%(&¦>0­,/B02B-(P7AI'J¬­20!46>Q(+Ld,6B-I'*G7,6H'>L5B%(¯024Ë7A,/7@-B-(B-,/*Í2J†,6H+0%C>>@
L(x˜0%C l 6= l 6= l CJ]02L H = H = H £Î¦>@%A0 .%" D "M°xLQLÊL,6¦2*-,/ÎH+0¥¶G@-(É7AB->L
1
2
3
D
1
D
2
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L('¯,6*7(+:%('7A02B-¡H+>J†0]B%,6LQ7AÍ2*-H',<0¡¦2(+>J†§'7,6H+0†('*7A(…¦>(+À2,6B-(=(…:>*7A
f gH H [%5MB
:N m C + CKC + 0Qq H ( ( pq ( [ ( C *I 0KM "
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ij
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j
X
g¯k − γ(φ
k=i
γ(φR )
R)
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„* O-J¡('†¦>(+2:27(+*-H',<024 0Qq H ( $( Yq ( [ ( C *I 0KM "
024Ë7A,/7@-B-(=>Q7>J¡§\7AQ,6H10“C H C§B-('Î*%,6B0ZH+2J¡ZLQ(+*-B-v0B-,/LQ7DÍ!*-H+,60¦>(+>J†§'7,/H10]J†(+B-,/B0
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=
Z
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Pi
dh0 ;
P00
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›“@-R%LQ7A,Ë7A@-,/*-B- dh CB-('Î-*-,6B-Z*-0](+¶G@02E+F2 .%" * C(H'>*-LQ,6B-('A02*-B%Z0¦>A01­,6B-02B-(02]46>*-¦2]B0
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0
0
i
HiO
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dW
=
gi0
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dC
=
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g
dh.
gi0
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02†46>*%¦>„B0¡4/,6*-U0B-(S:-Q@-J¡%C0¡024Ë7A,Ë7A@-B-(Ÿ2Q7A2J¡§\7A,/H10:“B-(TLQ(+•H1024/H+@-4602B0„B-,6Q('7D0!J¡('*G7(
0:02?7A,6}B0†LQ(+¦>@%,6*7A(T( “:-Q(+LQLAF2 0
i
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Ci
HiO
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1
= 0
ḡi
Z
HiO =
Pi
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gdh.
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P0
Ci
,
ḡi0
C £Î¦2@-A0 %. "/M "
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T/P traço 102 - lon(289,953°; 290,010°) e lat(-4,399°; -4,280°)
89
88
88
87
87
86
86
85
85
84
84
83
83
82
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Rio Solimões
(m)
T/P traço 165 - lon(294,572°; 294,648°) e lat(-2,629°; -2,450°)
46
46
45
45
44
44
43
43
42
42
41
41
40
40
39
39
38
38
37
37
36
36
35
34
35
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1.993
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Rio Solimões
(m)
T/P traço 254 - lon(295,207°; 295,277°) e lat(-3,280°; -3,120°)
40
40
39
39
38
38
37
37
36
36
35
35
34
34
33
33
32
32
31
31
30
30
29
29
28
27
26
28
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1.993
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27
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Rio Solimões
(m)
T/P traço 241 - lon(296,887°; 297,148°) e lat(-4,097°; -3,380°)
27
27
26
26
25
25
24
24
23
23
22
22
21
21
20
20
19
19
18
18
17
17
16
16
15
14
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1.993
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Rio Solimões
(m)
T/P traço 076 - lon(298,257°; 298,332°) e lat(-3,90°; -3,72°)
23
23
22
22
21
21
20
20
19
19
18
18
17
17
16
16
15
15
14
14
13
13
12
12
11
11
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10
9
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10
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Rio Amazonas
(m)
T/P traço 063 - lon(299,985°; 300,077°) e lat(-3,339°; -3,117°)
13
13
12
12
11
11
10
10
9
9
8
8
7
7
6
6
5
5
4
4
3
3
2
2
1
1
0
0
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-1
-2
1.993
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T/P traço 152 - lon(300,899°; 300,944°) e lat(-3,270°; -3,180°)
8
(m)
7
8
7
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one measure per cycle
6
6
5
5
4
4
3
3
2
2
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T/P traço 139 - lon(303,091°; 303,117°) e lat(-2,580°; -2,550°)
-1
-2
-2
-3
-3
-4
-4
-5
-5
-6
-6
-7
-7
-8
-8
(m)
-1
-9
-10
-9
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Rio Amazonas
(m)
T/P traço 228 - lon(303,500°; 303,528°) e lat(-2,580°; -2,503°)
-2
-2
-3
-3
-4
-4
-5
-5
-6
-6
-7
-7
-8
-8
-9
-9
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T/P traço 215 - lon(306,114°; 306,201°) e lat(-2,059°; -1,850°)
-14
-14
-15
-15
-16
-16
-17
-17
-18
-18
-19
-19
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1.993
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