A linear theory is developed for drift waves with frequencies between the electron and ion bounce frequencies in a collisionless plasma confined to a magnetic quadrupole. An integro-differential eigenvalue equation which allows for large ion-Larmor-radius effects, temperature gradients, and radial variation of the eigenfunction is derived from the linearized Vlasov equation. The fundamental drift mode observed in the UMIST steady-state quadrupole has one full wavelength around a closed magnetic field line with the electrostatic wave potential antisymmetric about the maximum field points. By numerical solution of the eigenvalue equation, the theoretical dispersion curve for this mode is extended to short wavelengths and the variation of the corresponding potential along a field line is determined.