1908-9.] 
On Lagrange’s Equations of Motion. 
327 
XIX. — On Lagrange’s Equations of Motion, and on Elementary 
Solutions of G-yrostatic Problems. By Professor Andrew 
Gray, F.R.S. 
(MS. received January 12, 1909. Read February 15, 1909.) 
1. 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. 
2. Long before Hertz’s book appeared, attention had been called to the 
subject; for instance, Ferrers pointed out in the Quarterly Journal of 
Mathematics , vol. xii. (1871-73), that in the case of a hoop rolling on a 
horizontal plane, while the equation for the inclination of the hoop to the 
vertical could be obtained from the expressions for the kinetic and potential 
energies by Lagrange’s method, that method failed to give the equations 
corresponding to the other co-ordinates. Erroneous solutions of the hoop 
problem have, however, since that time been published by more than one 
writer who had not perceived the fact noticed by Ferrers. An oversight 
of this kind in a solution of this problem given by another mathematician 
seems to have led Appell to his theorem ( Comptes Rendus , 1899) by which 
the equations of motion for holonomous and non-holonomous systems alike 
are obtained by what has since been called the “ kinetic energy of the 
accelerations of the system.” 
3. When Lagrange’s equations are deduced from the principle of least 
action, or by means of Hamilton’s characteristic function, the effect of the 
nature of the connections of the system is left more or less obscure. Some 
observations were made by Hertz on the subject, but these were far from 
conclusive, and it was first shown in 1896 by Holder ( Gott . Nadir., 1896), 
