Triply orthogonal coordinate systems

The properties of conformal transformations developed in the previous paper (Hotine 1965) are used to construct triply orthogonal coordinate systems from a scalar N. Classical doctrine on the subject requires that N should satisfy a third-order partial differential equation known as the Darboux equation, whereas no such restriction seems necessary in the conformal approach. If the Darboux equation is not a condition for triple orthogonality, but a property of the space, as this investigation suggests may be the case, then the effect on the geometry of the gravitation field would be considerable. It is accordingly important to resolve the question. The properties of triply orthogonal systems are worked out in some detail and are valid in all cases where a triply orthogonal system exists, whether or not the Darboux equation is a necessary condition. Some formulae in the differential geometry of curves and surfaces are required which are not to be found in the standard literature. For this reason, as well as to provide a clear statement of the conventions adopted, it has been necessary to provide a condensed, but connected, account of this subject in an appendix. References to the Appendix are in the form (10A). Inevitably in a connected account, some well-known relations are recovered, but they are usually arrived at by a new and shorter route and do serve to illustrate the method.