Earth’s figure and gravitation - Earth Dynamics: Deformations and Oscillations of the Rotating Earth - page 350

Equation (5.141) for the internal equipotential surfaces provides the expansions

1

m ( n

R n − 1 ∂ R

1 1

r n − 1
0

r 0 ∂ R 1
∂ r 0 ∂ r 0 =

− − 2) R 1 −

+ m 2 r 0 ∂ R 2

∂ r 0 R 1

( n − 1)

r 0 ∂

∂ r 0 −

− ( n − 2) R 2

2

R 1

( n − 1)( n − 2)

+ +··· , (5.152)

2

R − n + 2

− r − n + 2 /( n −

⎩

R

− 2), n 2,

dr
( r ) n − 1 =

ln R − ln r 0 , n = 2,

r 0

mR 1 + m 2 R 2 −

R 1

1
r n − 2
0

n − 1

= +···

, (5.153)

2

(1/ R 2 ) ( ∂ R /∂θ ) 2
∂ R /∂ r 0

1
R n − 1

1 +

1

m nR 1 +

1
r n − 1

r 0 ∂ R 1
∂ r 0

= −

0

+ m 2 ⎩ r 0 ∂ R 1

2

∂ r 0 R 1

− r 0 ∂ R 2

( n + 1)

r 0 ∂

∂ r 0 + ∂ r 0 2

2 ⎭ +···

⎦ ,

∂ R 1
∂θ

n ( n + 1)

R 1 + − nR 2 + (5.154)

2

for the integrands in (5.151), correct to second order. Only the first-order term in
the expansion (5.153) is needed. To second order, the scheme of equations (5.151)
contributes relations only for n =−

1,0,2 and 4. These may be found by substitut-
ing the functional forms (5.18) and (5.74) for R 1 and R 2 , with f and κ dependent
on r 0 , into the expansions (5.152) through (5.154) and carrying out the integrations
in (5.151) using the identities (5.41) and (5.49), as well as the orthogonality among
Legendre polynomials.
In stating the results, it is convenient to introduce the mean density within the
volume V i , bounded by the equipotential U i . The mass enclosed by U i is