In 1958, Courant and Snyder analyzed the problem of alternating-gradient beam transport and treated a model without focusing gaps or space charge. Recently we revisited their work and found the exact solution for matched-beam envelopes in a linear quadrupole lattice [O.A. Anderson and L.L. LoDestro, Phys. Rev. ST Accel. Beams, 2009]. We extend that work here to include the effect of gaps. We derive the exact envelopes and show results for various field strengths, occupancies eta, and gap-length ratios. We find the peak envelope excursion. It has a broad minimum as a function of the phase advance sigma (typically around 34o) over which it varies less than +-1percent. The phase-advance numbers also change little over the full range of gap ratios. However, the required field strengths vary appreciably. In the second stable band, the higher field strength necessitated by the lower occupancy accentuates the remarkable compression effect predicted for the FD (gapless) model.