Improved formulas are developed for the optimum tuning of damped vibration absorbers whose motion may or may not be in-line with the motion axis of the main mass. J. Ormondroyd and J. P. Den Hartog (1928), studied a class of dynamic vibration absorbers whose motion is in-line with that of the main mass and reported the existence of optimum tuning and damping values for the absorber mass along with supporting numerical results. An asymptotic solution for the optimum damping of J. Ormondroyd's and J. P. Den Hartog's absorber was presented by J. E. Brock (1946); however, his solution breaks down as the mass ratio increases. In this paper, an exact solution for the optimum damping of J. Ormondroyd's and J.P. Den Hartog's absorber is given along with an analysis of vibration absorbers whose motion is not in-line with the motion axis of the main mass. This work was made possible by the use of a symbolic algebra program which was not available to the earlier investigators.