  A rigorous analytical theory of parametric excitation under the longitudinal
pumping has been developed for the surface magnetostatic modes of a long
elliptic ferrite cylinder magnetized along its axis with regard for the
boundary conditions at the surface of the cylinder.It is shown that a pair of
frequency-degenerated counter-propagating surface modes at half the pumping
frequency can be parametrically excited, and the expressions for the
corresponding parametric excitation threshold have been derived. The threshold
demonstrates a strong dependence on the mode number and elliptic cylinder's
aspect ratio and tends from above for the large aspect ratio to the value
deduced on the basis of the plane-wave analysis.The simple analytical relation
between the ratio of axes of the high-frequency magnetization polarization
ellipse of excited surface magnetostatic oscillations and the parametric
excitation threshold is obtained, discussed, and graphically illustrated.
