It is well known that the theory of the small resonator provided with a neck communicating with the open air depends upon the assumption that the air within the neck may be treated as an incompressible fluid. In § 2 of the present paper the theory is extended to include generally necks of variable cross-section. Again, Rayleigh develops three theories depending upon the assumptions made as to the velocity-distribution of the air at the open end of a resonator. It is shown in § 3 that his theory, which assumes a variable velocity-distribution over the open end, is erroneous, and a theory which appears to be correct is put forward. The theory of the neck of variable cross-section is applied to the Helmholtz resonator in § 4. In § 5 the theory of an open end of rectangular cross-section is considered. In § 6 the general theory of resonators is dealt with. It is shown that impedance principles require modification for certain types of resonator, although they hold for most of the common types.