It is now well known that Lagrange's equations of motion for a system of connected particles are not applicable to certain cases of motion—for example, to a rigid sphere rolling without sliding on a given surface. In his Principien der Mechanik, Hertz has referred at considerable length to this subject, and has applied the adjective “holonomous “to those systems to which the equations are applicable, and has called all others “non-holonomous.” These adjectives correspond to distinct characteristics of the systems as regards the constraints to which they are subject. Holonomous systems are those in which the constraints are expressed, or can be expressed, by finite equations; in non-holonomous systems, on the other hand, these conditions, or some of them at least, are expressed by differential relations, which do not fulfil the conditions of integrability.