An instability in synchronous machinery which cannot be assigned to variations in the load torque or driving torque is frequently experienced. This instability was shown by Hopkinson to be due to a negative damping coefficient which can, under certain circumstances, exceed the positive damping inherent in the machine and apparatus to which it may be connected. If this condition be achieved then any displacement from the normal running position will result in a sinusoidal vibration (about this position), increasing exponentially, until the machine either falls out of step, or (the coefficient of negative damping becoming modified by the effects of increasing displacement) settles down to a steady oscillation.In the first part of this paper the work of Rosenberg and Kapp is recapitulated to demonstrate the possibility of harmonic vibrations (about the normal running position) of alternators working in parallel. An expression is next derived for the damping coefficient of a synchronous machine in which the cross-magnetizing and demagnetizing coefficients are equal, and in which no damping grids are fitted. Some of the effects of armature reaction are ignored in this case. The conditions for instability are investigated.The theory is next extended to the case of a machine having unequal coefficients of cross-magnetization and demagnetization, and the effects of armature reaction are considered in greater detail. An expression for the damping torque is derived, and the conditions for negative damping are again investigated.Damping grids are then assumed to be fitted to the poles and in the interpolar spaces of the unsymmetrical machine last considered, and an expression for the damping torque is arrived at in which the armature resistance is as a first approximation neglected. This expression is compared with the torque equation of an induction motor to show how, upon analogy, the resistance of the grids for a damper of given performance may be calculated. An expression is also derived by which the negative damping due to the armature resistance may be assessed.In the section of the paper devoted to the consideration of experimental work, measurements are described which were undertaken to confirm the formulae obtained from mathematical analysis, and it is shown that fairly good agreement can be obtained between the measured and calculated magnitudes.The Appendices are concerned with the further elucidation of certain points in the mathematical treatment, and with the consideration of the positive damping inherent in synchronous machines, and in certain types of apparatus to which they may be mechanically connected.