The transmission of extra low frequency transverse waves with time periods 1 to 600 seconds propagating along the geomagnetic field from the msgnetospheric boundary to the surface of the Earth is considered. A nearly exact analytical solution for the propagation through the entire highly inhomogeneous path is obtained. For typical magnetospheric conditions pronounced maxima of transmission are found at time periods in the vicinity of 30,47, 70, 136 and 218 s. More moderate resonances are seen at 37, 39, 50, 77 and 99s. Closely spaced small resonances are found between 7 and 36 s and between 54 and 64 s and two extremely small ones are seen at 1.5 and 4 s. Giant resonances are seen at 168, 31 1 and 542 s.