In 1970 a new world beckoned, the realm of resonance, with prospects of fresh and fertile fields of research. A satellite experiences resonance when longitudinal variations in gravity cause changes in the orbit that build up continually, day after day and month after month. Orbital changes that are basically very small then magnify themselves until they are large enough to be accurately determined: thus resonance creates a powerful technique for measuring the gravity field.

In earlier chapters the Earth's gravity has been taken to be composed of a series of zonal harmonics dependent only on latitude, and independent of longitude. This is an over-simplification, because in reality gravity varies with longitude: the variations are small, but detectable. The zonal harmonics discussed in previous chapters can be regarded as longitude-averaged, and each of them needs to be supplemented by a teeming family of harmonics that are dependent on longitude as well as latitude,‘tesseral harmonics’ as they are called, after the tesserae of varied shapes in a Roman mosaic floor.

The variation of a tesseral harmonic with longitude is specified by its order. A tesseral harmonic of order 15 gives rise to 15 undulations as you go round the equator (or any other line of latitude), as shown in Fig. 5.1. The symbol m is used to denote the order of a tesseral harmonic: it is helpful to think of m as specifying the variations between one meridian and another. (The zonal harmonics, being independent of longitude, are tesseral harmonics of order zero.)