Appendix A

The segment $P\tau$ in $SPO^\prime$ is given as (see Fig. 1)

$$P \tau = OP \ [A (B^{\prime}/B) + (1 - A^{2})^{1/2}],$$

where

$$B = (OP/OT)A, B^{\prime} = (1 - B^{2})^{1/2},$$

$$A = \xi \eta^{\prime} - \xi^{\prime} \eta,$$

$$\eta = \sin \ E = SE/PE, \eta^{\prime} = \cos \ E,$$

$$\xi = abc + a^{\prime} b^{\prime} c + a^{\prime} b c^{\prime} - a b^{\prime} c^{\prime},$$

$$\xi^{\prime } = (1 - \xi^{2})^{1/2}.$$

And

$$a = OQ/OP, a^{\prime} = (1 - a^{2})^{1/2},$$

$$b = PS/O^{\prime} P, b^{\prime} = (1 - b^{2})^{1/2},$$

$$c = PQ/O^{\prime} P, C^{\prime} = (1 - c^{2})^{1/2}.$$

Similarly the segments such as $P\tau^\prime$ in $O^\prime$ PW are given by,

$$P\tau^{\prime} = OP [\mu(s^{\prime}/s) + (1 - \mu^2)],$$

where

$$\mu = \nu^{\prime} \Delta + \Delta^{\prime} \nu,$$

$$s = (OP/OT) \mu, \ s^{\prime} = (1 - s^{2})^{1/2},$$

$$\nu = WE^{\prime}/PE^{\prime}, \ \nu^{\prime} = (1 - \nu^{2})^{1/2},$$

$$PE^{\prime 2} = PW^2 + WE^{\prime 2}, \ \Delta = \xi (1 -2 \delta^2) - 2 \xi^{\prime} \delta \delta^{\prime},$$

$$\Delta^{\prime} = (1 - \Delta^2)^{1/2}, \ \delta = SO^{\prime}/PO^\prime,\ \delta^{\prime} = (1 - \delta^2)^{1/2}.$$