This paper is the second of a series of three. The first dealt with the steady forced harmonic motion of a simple oscillator with one degree of freedom when “ hysteretic damping ” is present; by this it is meant that a damping force exists which is in phase with velocity but whose magnitude is proportional to displacement. The present paper is devoted to the formulation of a general theory of small hysteretically damped vibration in which the existence is postulated of damping forces which act between pairs of points such that they are in phase with the relative velocities and their magnitudes are proportional to relative displacements. The theory is presented in general terms and use is made of Lagrange’s equations with a new type of “ dissipation function ”; this is comparable with that of Rayleigh for viscous damping. It is shown that this theory generally leads to simpler algebra than does that of viscous damping. Moreover, it will be shown in a later paper that the nature of forced and damped oscillation may be thought of in terms of modes more simply than is the case with the more familiar viscous damping.