content_image/368489.jpg content_image/368490.jpg content_image/368488.jpg content_image/368486.jpg content_image/368485.jpg content_image/368483.jpg content_image/368484.jpg content_image/368482.jpg
To relate t to s , we identify 

<img src='content_image/368489.jpg'>

<img src='content_image/368490.jpg'>

Solving the quadratic equation for t versus s , we obtain the minimizing function t ∗ ( s ), which obviously obeys the boundary condition 

<img src='content_image/368488.jpg'>

The symmetry s → 1 /s plays apparently an important role. It guarantees that the points −∞ , − 1, 0, +1, + ∞ are mapped onto themselves under the reparametrization s → t ∗ ( s ): t ∗ ( −∞ ) = −∞ , t ∗ ( − 1) = − 1, t ∗ (0) = 0, t ∗ (+1) = +1, t ∗ (+ ∞ ) = + ∞ . 

It is convenient to invert Eq. (28) using the Jacobi elliptic functions. Inverting Eq. (28a), we get 

The function sn has a nice trigonometric expansion in the parameter (nome) (Ref. [11], 8.146.1) 

giving 

This formula is applicable for − R/ 2 T < t < + R/ 2 T , when − √ µ < s < + √ µ . 

We can proceed in the same way with Eq. (28b), whose inverse is 

<img src='content_image/368486.jpg'>

for 

<img src='content_image/368485.jpg'>

<img src='content_image/368483.jpg'>

<img src='content_image/368484.jpg'>

Using the addition formula ([11], 8.156.1) and the reduction of sn( x, 0) and cn( x, 0) to sin x and cos x , we obtain 

<img src='content_image/368482.jpg'>

