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An axisymmetric transversely isotropic material is considered. A complete solution in terms of two retarded potential functions for the wave equations of general elastodynamics problems in transversely isotropic media has been presented in literature, which reduced to one for torsionless axisymmetric case. In the cylindrical coordinate system , the body force in the torsionless axisymmetric case given there is at most in the form of . In this paper, we show that by introducing another potential function, which is directly obtained from the body force in radial direction, a new complete solution is introduced for more general torsionless axisymmetric case, where the body force may be as general as . The new solution is degenerated for the elastostatic case, and in addition it is degenerated for the torsionless axisymmetric elastodynamics and elastostatics problem of isotropic materials. The solution for elastostatics problem of isotropic material is the same as extended Love's potential function.