  We investigate superconductivity in a grand canonical ensemble with {\it
fixed number parity} (even or odd). In the low temperature limit we find small
corrections to the BCS gap equation and energy spectrum $E(k)$. The even-odd
free energy difference in the same limit decreases linearly with temperature,
in accordance with the behavior observed experimentally and previously arrived
at from a quasiparticle model. The theory yields deviations from the BCS
predictions for the specific heat, ultrasound attenuation, NSR relaxation rate,
and electromagnetic absorption.
