156 
[195 
195. 
REPOET ON THE RECENT PROGRESS OF THEORETICAL 
DYNAMICS. 
[From the Report of the British Association for the Advancement of Science, 1857, 
pp. 1—42.] 
The object of the Mécanique Analytique of Lagrange is described by the author 
in the “ Avertissement ” to the first edition as follows :—“ On a déjà plusieurs traités 
de mécanique, mais le plan de celui-ci est entièrement neuf. Je me suis proposé de 
réduire la théorie de cette science et l’art de résoudre tous les problèmes qui s’y 
rapportent à des formules générales dont le simple développement donne toutes les 
équations nécessaires pour la solution de chaque problème.” And the intention is 
carried out ; the principle of virtual velocities furnishes the general formulae for the 
solution of statical problems, and d’Alembert’s principle then leads to the general 
formulae for the solution of dynamical problems. The general theory of statics would 
seem to admit of less ulterior development ; but as regards dynamics, the formulae of 
the first edition of the Mécanique Analytique have been the foundation of a series of 
profound and interesting researches constituting the science of analytical dynamics. The 
present report is designed to give, so far as I am able, a survey of these researches ; 
there will be found at the end a list, in chronological order, of the works and memoirs 
referred to, and I shall in the course of the report preserve as far as possible the 
like chronological order. It is proper to remark that I confine myself to the general 
theories of dynamics. There are various special problems of great generality, and 
susceptible of the most varied and extensive developments, such for instance as the 
problem of the motion of a single particle (which includes as particular cases the 
problem of central forces, that of two fixed centres, and that of the motion of a conical 
pendulum, either with or without regard to the motion of the earth round its axis), 
the problem of three bodies, and the problem of the rotation of a solid body about 
a fixed point. But a detailed account of the researches of geometers in relation to 
these special problems would properly form the subject of a separate report, and it 
is not my intention to enter upon them otherwise than incidentally, so far as it may 
appear desirable to do so. One problem, however, included in the first of the above-