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The particles in an accelerator interact with one another by electromagnetic forces and are held together by external focusing forces. Such a many-body system has a large number of transverse modes of oscillation (plasma oscillations) that can be excited at characteristic frequencies by errors in the external guide field. In Part I we examine one mode of oscillation in detail, namely the quadrupole mode that is excited in uniformly charged beams by gradient errors. We derive self-consistent equations of motion for the beam envelope and solve these equations for the case in which the space-charge force is much less than the external focusing force, i.e., for strong-focusing synchrotrons. We find that the resonance intensity is shifted from the value predicted by the usual transverse incoherent space-charge limit; moreover, because the space-charge force depends on the shape and size of the beam, the beam growth in always limited. For gradient errors of the magnitude normally present in strong-focusing synchrotrons, the increase in beam size is small provided the beam parameters are properly chosen; otherwise the growth may be large. Thus gradient errors need not impose a limit on the number of particles that can be accelerated. In Part II we examine …