The purpose of this thesis is to investigate the properties and wave velocities for an axially symmetric medium. The investigation consists of four parts. In the first part, the physical properties of the medium are defined. Then the stress-strain relations for the case under consideration are obtained from the general case. This is done by imposing the condition of symmetry on the strain energy function. Next the measurable constants are round in terms or the natural constants. This is done by applying simple extensions and shears to the material. After the measurable constants are determined in terms or the natural constants, then the relationship is inverted and the natural constants are found in terms or the measurable constants. Some elastic constants are then determined for an arbitrary direction, as it is not likely that the stresses will always be imposed along an axis of symmetry. Following this major part, the equations of motion for a vibrating medium are determined in terms or the natural constants by substituting the stress-strain relations. This is very straightforward, but must be done. The final step is made by finding the velocities of propagation of the waves by using the equations of motion. This is done by assuming a solution and substituting into the equations of motion. From these equations, a cubic equation defining the three principal velocities arises. The solution of this cubic equation is the culmination of this investigation. / M.S.