REPORTS 
ON 
THE STATE OF SCIENCE. 
Report on the Recent Progress of Theoretical Dynamics. 
By A. CaYuery. 
Tue object of the ‘Mécanique Analytique’ of Lagrange is described by the 
author in the ‘ Avertissement’ to the first edition as follows :—“ On a deja 
plusieurs traités de mécanique, mais le plan de celui-ci est entiérement neuf. 
Je me suis proposé de réduire la théorie de cette science et l’art de résoudre 
tous les problémes quis’y rapportent a des formules générales dont le simple 
développement donne toutes les équations nécessaires pour la solution de 
chaque probléme.’”? And the intention is carried out; the principle of virtual 
velocities furnishes the general formulz for the solution of statical problems, 
and D’ Alembert’s principle then leads to the general formula for the solu- 
tion of dynamical problems. The general theory of statics would seem to 
admit of less ulterior development; but as regards dynamics, the formule 
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 dyna- 
mies. 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 
eases 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 desir- 
able to do so. One problem, however, included in the first of the above- 
mentioned special problems, I shall have frequent occasion to allude to: I 
mean the problem of the variation of the elements of a planet’s orbit, which 
has a close historical connexion with the general theories which form the 
“subject of this report. The so-called ideal coordinates of Hansen, and the 
ge of his method of integration in the planetary and lunar theories, 
have a bearing on the general subject, and might have been considered in 
the present report ; but on the whole I have considered it better not to do so. 
1. Lagrange, ‘ Mécanique Analytique,’ 1788.—The equations of. motion 
are obtained, as before mentioned, by means of the principle of virtual velo- 
1857. B 
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