We develop the linear theory of drift waves in a sheared quadrupole magnetic field configuration, with a view to applying the theory to the UMIST quadrupole GOLUX; shear can be introduced into this system by imposing a uniform longitudinal field. An eigenvalue equation is obtained, and appropriate sets of boundary conditions are proposed. The basic instability is due to the 'dissipative trapped electron' mechanism in both the simple and sheared quadrupole configurations, but the mode structure changes with shear; at sufficiently large values the mode adopts the Pearlstein-Berk (1969) form and may then be stabilized by shear damping. A novel prediction of the theory is that the 'private flux' region of the quadrupole, which without shear is completely stable, is destabilized at very small values of longitudinal field. We propose that the sheared quadrupole will form an excellent laboratory system for testing theories of drift waves developed for tokamak configurations.