Numerical calculations have been made on an IBM 7090 of the periods and particle motion with depth of the lowest order free spheroidal vibrations of the fundamental and first two higher modes with inner core shear velocity as a parameter. All calculations were made for a model with velocities according to Jeffreys and with densities obeying Bullen's model B. These calculations show that the effect of the inner core shear velocity on these periods is slight, except for those velocities at which the modes have periods near to the periods of core modes. This means that observations of long period free vibrations will probably not give much information about rigidity of the inner core. A mode of vibration having many of the properties of the mode suggested by Slichter has been discovered. In addition, it appears that the presence of a solid inner core brings into existence many core type modes with periods both shorter and longer than the original core mode predicted by Alterman, Jarosch, and Pekeris.