Earth deformations - Earth Dynamics: Deformations and Oscillations of the Rotating Earth - page 251

required for the calculation of k 4 . Thus,

h ∂ 2 h ∂ h ∂ f

1
2!

k 3 ∂
∂y

1
2

k 3 ∂
∂y

k 3 ∂ f
∂y ∂ r +

f =

∂ r + ∂ r +

h 2 ∂

2 f
∂ r +

2 f
∂ r ∂y +

2 f
∂y

1
2

2 hk 3 ∂ k 3 ∂ =

2

1
2

h 2 ∂

2 h hf

2 Df ∂

2 f
∂ r 2 +

h 2

2 f
∂ r ∂y

= +

r = r i +

h 2 f 2

h 3 fDf ∂

2 f
∂y

O h 4 + +

2

r = r i + h 2

2 Df ∂

h 3

2 f
∂ r ∂y +

2 f
∂y

O h 4 f ∂

2 D 2 f

= +

2

r = r i + h 2

h 3
2 Df Df y

O h 4 ,

2 D 2 f

= + (3.291)

again using (3.268) and the identity

2 f
∂ r ∂y +

2 f
∂y

∂

f i ∂

2 =

Df y . (3.292)

Again,

h ∂ 3 h ∂ h 2 ∂

2 f
∂ r 2 +

2 f
∂ r ∂y +

2 f
∂y

1
3!

k 3 ∂
∂y

1
6

k 3 ∂
∂y

2 hk 3 ∂ k 3 ∂

∂ r +

f =

∂ r +

2

r = r i +

h 3
6

∂
∂ r +

∂

2 f
∂ r 2 +

2 f
∂ r ∂y +

2 f
∂y

f ∂
∂y

2 f ∂ f 2 ∂ O h 4 =

2

6 D 3 f r = r i +

h 3

O h 4 , = (3.293)

using (3.270). From (3.290) with substitution from (3.289), (3.291) and (3.293),
we find that

k 4 = hf ( r i + h ,y i + k 3 )

hf

2 D 2 f

f y Df

h 3

h 2 Df = + + +

r = r i + D 3 f

h 4
6

3 3

O h 5 .

4 f y D 2 f 2 f 2

+ + +

y Df

+ 3 Df Df y (3.294)