Earth’s rotation: observations and theory - Earth Dynamics: Deformations and Oscillations of the Rotating Earth - page 298

valid for r > r . Thus, we can write that

r 3

Gx j

GM
r −

x j ρ( r ) d V ( r ) =− V

V

j

(4.45)

2 r 5 3 x i x j −

G

r 2

i

x i x j ρ( r ) d − δ V+··· .

V

In a reference co-ordinate system with origin at the centre of mass, by definition

x j ρ( r ) d V= 0. (4.46)

V

From expression (4.20) for the components of the inertia tensor, we find that

x i x j ρ( r ) d r 2

j ρ( r ) d

V=− I ij + δ V . (4.47)

V V

Again, from expression (4.20), the trace of the inertia tensor is

2

r 2 ρ( r ) d Tr( I ) = I 11 + I 22 + I 33 = V . (4.48)

V

Thus,

1
2 Tr( I )δ

x i x j ρ( r ) d

i

V=− I ij +

j .

(4.49)

V

r 2

i

In the expansion (4.45), this expression is multiplied by 3 x i x j − δ

j , and the second

term on the right side of (4.49) yields

2 Tr( I ) 3 x 1 −

r 2

1

r 2 3 x 2 − r 2 3 x 3 − + + = 0. (4.50)

Hence, the expansion (4.45) takes the form

GM
r

V ( r ) =−

2 r 5 I 11 x 2 +

2 x 1 I 22 x 3 + 2 x 2 I 33 x 1 + 2 x 3

G

x 3 − x 1 − x 2 − − + +

6 I 31 x 3 x 1 − 6 I 12 x 1 x 2 − 6 I 23 x 2 x 3 − +··· . (4.51)

Relations connecting the gravitational potential to the components of the inertia
tensor are referred to as MacCullagh's formula (MacCullagh, 1845). Converting to
spherical polar co-ordinates, and again using the identities (4.42), at the surface of