(12.202) | ||

(12.203) |

where

Here, use has been made of Equations (12.186) and (12.190). Furthermore [cf., Equations (12.159) and (12.160)],

Let , , , and in Equations (12.191)-(12.196). It follows that

(12.208) | ||

(12.209) |

where

Furthermore,

Substitution of Equations (12.211), (12.212), (12.217), and (12.218) into Equations (12.187) and (12.188) yields

where

(12.216) | ||

(12.217) |

Equations (12.219) and (12.220) can be combined to give

Moreover, it follows from Equations (12.188), (12.190), (12.209), (12.210), (12.215), and (12.216) that

We have already seen that the solution of Equation (12.205), subject to the boundary condition (12.204), is . It follows that the solution of Equation (12.223), subject to the boundary condition (12.225), is . Analogous arguments reveal that the solution of Equation (12.224), subject to the boundary condition (12.226), is . Hence, we deduce that the Laplace tidal equations, (12.186)-(12.188), are equivalent to the following set of equations:

where , , , , and are defined in Equations (12.211), (12.212), (12.217), (12.218), and (12.189), respectively.