Vibrations of free and clamped transversely isotropic plates with an internal flat circular slot are studied. According to the physical conditions, the normal stresses become uniform within the circular slot and displacements, and the shearing stresses are zero outside the circular slot. These are mixed boundary value problems which give rise to dual integral equations. Following Noble, the dual integral equations are reduced to Fredholm integral equations. Kernals of the integral equations are evaluated by the contour integration technique. The stress intensity factors are obtained for both cases and corresponding isotropic results are deduced. Numerical analysis is performed in the case of Beryl material. The relationship between the dimensionless frequency and the stress intensity factor is presented. The results thus obtained are discussed.