APPENDIX B

To derive equation (16), I take the derivative of both sides of equation (15) with respect to s and map the result to Cartesian coordinates, which yields

$${\partial \eta \over \partial x}{dx \over ds}+ {\partial \et... ...\partial z}{dz \over ds}= {m(x_0,z_0) \over m(x,z)}. \eqno(B.1)$$

The ray equation (6) gives

$$\begin{array} {lll} \displaystyle{dx \over ds} & = & \displa... ...& \displaystyle{\tau_z(x,z) \over m(x,z)} \end{array}\eqno(B.2)$$

Substituting these relations from equation (B.2) into equation (B.1) and multiplying both sizes by m(x,z) yields equation (16):

$$\tau_x\eta_x+\tau_z\eta_z=m(x_0,z_0). \eqno(B.3)$$

When (x,z) tends to (x₀,z₀), s tends to s₀ and $\eta$ tends to zero, which gives the initial condition.