The effect of trapped particles on the drift-cyclotron instability in a multipole machine has been considered. Dispersion equations have been obtained for this instability in a nonuniform collisionless plasma using a simplified approach which takes into consideration the magnetic field strength and curvature variation along field lines. The results show the existence of a branch of oscillations with frequency |ω|~Ω0-kv̄g- ω̄b, where Ω0, v̄g, and ω̄b are the average gyrofrequency, the drift velocity due to magnetic field gradient, and the ion "bounce" frequency, respectively, with k being the wave vector transverse to the magnetic field. This may explain the discrepancy between the experimentally observed frequency and the theoretical value obtained previously. This branch of the oscillation also shows a larger growth rate.