This paper describes the vibration analysis of a three-element dynamic vibration absorber. The absorber is of one and one-half degrees of freedom and consists of an absorber mass, an absorber spring and an elastically connected damper. The effectiveness of the absorber in reducing the resonance peaks is greater than that of the two-element conventional absorber. The maximum force acting on the absorber spring is comparatively small for the effect of the elastically connected damper. When the main system has no damping, there are three effective fixed points on the amplitude curve of the main mass. Formulae for optimum tuning and damping are derived from a new thery of three fixed points. In the damped main system, the optimum values of the absorber are obtained through the numerical analysis using a Lagrange multiplier. The numerical results are given by the convenient empirical formulae.