Earth’s rotation: observations and theory - Earth Dynamics: Deformations and Oscillations of the Rotating Earth - page 305

Now,

sin ( β − γ )
cos (β − γ)

tan (β − γ) =

sinβcosγ − cosβsinγ

=

cosβcosγ + sinβsinγ

tanβ − tanγ

=

tanβtanγ ,

(4.91)

1 +

and hence,

tanβ − tanγ Ω

tanβtanγ = Ω+ σ 0 ) tanβ,

(4.92)

1 + (

on choosing the positive root. This gives

1

Ω Ω

Ω+ σ 0 tanβ Ω+ σ 0 tan 2

tanβ − tanγ + β = (4.93)

or

− Ω 1

tan γ
tanβ =

σ 0
σ 0 +Ω

Ω+ σ 0

β =

β .

(4.94)

+ Ω

sec 2

tan 2

1

Ω+ σ 0

The quantities β, σ 0 , are all constants of the motion, therefore γ is also a con-

stant of the motion. The angular velocity vector ω then describes a small cone (the
herpolhode) about the invariable axis L with semi-apex angle γ.
Of interest is the rate of progression of ω about L .Letξ measure the angular
position of ω in its path about L . In time δ t the increase in ξ is δξ. The component
of ω normal to L is

Ω

sinγ. This acts as a radius vector moving through the angle
δξ in time δ t to produce the change δ ω in time δ t . Then,

| ω |

L × ω

δ ω = δξ | ω | sinγ = δξ · (4.95)

| L |

and the time derivatives of ω and ξ are related by

L × ω

˙

ω

=

ξ . (4.96)

| L |

We also have that

ω =Ω m 1 e 1 +Ω m 2 e 2 . (4.97)