Introduction and theoretical background - Earth Dynamics: Deformations and Oscillations of the Rotating Earth - page 90

In the odd case, the supporting polynomials are seventh degree. With some
experimentation, they can be expressed in terms of the canonical Hermite cubics
by

0 (ζ i )

z
z i

1
z i ζ i ( z i ) ψ

i ( z ) 0 (ζ i )

η = η − , (1.585)

0 (ζ i + 1 )

z
z i + 1

1
z i + 1 ζ i + 1 ( z i + 1 ) ψ

L L L

η

i + 1 ( z )

= η

0 (ζ i + 1 )

− , (1.586)

z
z i

1
ζ i ( z i ) ψ

R R

ψ

i ( z )

=

0 (ζ i ),

(1.587)

z
z i + 1

1
ζ i + 1 ( z i + 1 ) ψ

L L

ψ

i + 1 ( z )

=

0 (ζ i + 1 ).

(1.588)

Their derivatives are

1
z i η

z

η R

R

z i ζ i ( z )η 0 (ζ i )

( z ) =

0 (ζ i )

+

i

z 2

1
z i ζ i ( z i ) ψ

R

z i ψ 0 (ζ i ),

−

0 (ζ i )

− (1.589)

1
z i + 1 η

z

η L

L

z i + 1 ζ i + 1 ( z )η 0 (ζ i + 1 )

i + 1 ( z )

=

0 (ζ i + 1 )

+

z 2 1

z i + 1 ζ i + 1 ( z i + 1 ) ψ

L

z i + 1 ψ 0 (ζ i + 1 ),

−

0 (ζ i + 1 )

− (1.590)

and

z 2 1

z i ζ i ( z i ) ψ

ψ R

z i ψ 0 (ζ i )

R

( z ) = +

0 (ζ i ),

(1.591)

i

z 2 1

z i + 1 ζ i + 1 ( z i + 1 ) ψ

ψ L

z i + 1 ψ 0 (ζ i + 1 )

L

i + 1 ( z )

= +

0 (ζ i + 1 ).

(1.592)

0, is an exception.
In the even case, the derivative vanishes at the equator, and in the odd case, the
function vanishes there. Thus lower order polynomials can be used for the approx-
imating function. In the even case,

The interval ( z 1 , z 2 ) beginning at the equator, with z = z 1 =

c 2 z 2 c 4 z 4 s ( z ) = c 0 + + (1.593)