The interaction between a beam of large-orbit axis encircling electrons and a waveguide is studied by linearizing the fluid equations. A dispersion relation is derived for an annular layer of relativistic electrons of small thickness guided by a uniform magnetic field along the waveguide axis. It is shown in the general case that the dispersion relation couples the TE and TM modes of the waveguide at the two resonances omega -kz upsilon 0z=(l+or-1) omega c and omega -kz upsilon 0z=l omega c. The resonance at omega -kz upsilon 0z=(l+or-1) omega c which is associated with the finite velocity shear (( delta nu 0z/ delta rho ) not=0) and finite kz is shown to give the dominant contribution to the dispersion relation.