9 Natural Measuring-Rods and Clocks
83
sin α =
tan α
√
1 + tan 2 α
, cos α =
1
√
1 + tan 2 α
, hence
sin 4α =
1
√
1 + tan 2 α
4
tan α
√
1 + tan 2 α
− 8
tan
3
α
(
√
1 + tan 2 α) 3
= 4
tan α
1 + tan 2 α
1 − 2
tan
2
α
1 + tan 2 α
= 4
tan α
1 + tan 2 α
1 + tan
2
α − 2 tan
2
α
1 + tan 2 α
,
sin 4α = 4 tan α
1 − tan
2
α
(1 + tan 2 α) 2
and thus with α = arctan e
x
sin(q
I
o ) = sin(4 arctan e
x
) = 4e
x 1 − e
2x
(1 + e 2x ) 2 .
On the other hand, a repeated differentiation gives us
∂
∂x
(4 arctan e
x
) = 4
e
x
1 + e 2x ,
∂
2
∂x 2 (4 arctan e
x
) = 4
∂
∂x
e
x
1 + e 2x = 4
e
x
(1 + e
2x
) − e
x 2e
2x
(1 + e 2x ) 2
and thus
∂
2
∂x 2 q
I
o =
∂
2
∂x 2 (4 arctan e
x
) = 4e
x 1 − e
2x
(1 + e 2x ) 2 .
Hence, the time-independent field q
I
o = q
I
o (x) does in fact fulfil Eq. (95),
∂
2 q
I
o
∂x 2 = sin q
I
o .
The function (98a) describes a line form immediately observable in the continuum.
What does this line look geometrically? As one can see from (98) the dislocation
line passes for x −→ −∞ into the position of equilibrium q = 0 and changes in
the vicinity of x = 0 into the neighbouring position of equilibrium q = a , into
which it passes for x −→ +∞. Here, the changing into the neighbouring position
of equilibrium in the vicinity of x = 0 takes place in a relatively small range of the
order λ o ; see Fig. 9.1. Those types of kinks can be found in large numbers inside of