Earth deformations - Earth Dynamics: Deformations and Oscillations of the Rotating Earth - page 250

using the expression (3.268) for D 2 f , and recognising the identity

2 f
∂ r ∂y +

2 f
∂y

∂

f i ∂

2 =

Df y . (3.287)

For the term following (3.286), we have

h 3

1
3! 2 ∂

k 2

2 ∂ ∂ r +

f

∂y

h h 2 ∂

2 f
∂ r 2 +

2 f
∂ r ∂y +

2 f
∂y

1
24 2 ∂

k 2

2 ∂

2 hk 2 ∂ k 2 ∂

=

∂ r +

2

∂y

O h 4 r = r i

h 3
48

∂
∂ r +

∂

2 f
∂ r 2 +

2 f
∂ r ∂y +

2 f
∂y

f ∂
∂y

2 f ∂ f 2 ∂ = +

2

O h 4 r = r i

h 3
48

∂

3 f
∂ r 3 +

3 f
∂ r 2

3 f
∂ r ∂y

3 f
∂y

3 f ∂ 3 f 2 ∂ f 3 ∂

=

∂y +

2 +

+

3

48 D 3 f + O h 4 r = r i

h 3

= (3.288)

upon substitution from (3.270). From (3.284), with substitution from (3.283),
(3.286) and (3.288), we find that

hf r i

hf

8 D 2 f

2 f y Df

h 2
2 Df

h 3 h

2 ,y i

k 2
2

k 3 = + + = + + +

6 Df Df y r = r i +

48 D 3 f

h 4

O h 5 . 3 f y D 2 f + + + (3.289)

In turn, this expression for k 3 can be used to develop the expansion,

h ∂ 2

h ∂ f k 3 ∂ f

1
2!

k 3 ∂
∂y

f ( r i + h ,y i + k 3 ) = f ( r i ,y i ) +

∂ r + ∂y + ∂ r +

f

h ∂ 3

1
3!

k 3 ∂
∂y

+

∂ r +

f +··· (3.290)