PROGRESS IN NINETEENTH CENTURY 35 



and potential energy from dynamics altogether postulating a uni- 

 verse of concealed motions such as Helmholtz (1884) had treated 

 in his theory of cyclic systems, and Kelvin had conceived in his 

 adynamic gyrostatic ether (1890). In fact, the introduction of con- 

 cealed systems and of ordered molecular motions by Helmholtz and 

 Boltzmann has proved most potent in justifying the Lagrangian 

 dynamics in its application to the actual motions of nature. 



The specific contributions of the first rank which dynamics owes 

 to the last century, engrossed as it was with the applications of the 

 subject, or with its mathematical difficulties, are not numerous. 

 In chronological order we recall naturally the statics (1804) and 

 the rotational dynamics (1834) of Poinsot, all in their geometrical 

 character so surprisingly distinct from the contemporary dynamics 

 of Lagrange and Laplace. We further recall Gauss's principle of 

 least constraint (1829), but little used, though often in its appli- 

 cations superior to the method of displacement; Hamilton's prin- 

 ciple of varying action (1834) and his characteristic function (1834, 

 1835), the former obtainable by an easy transition from D'Alem- 

 bert's principle and by contrast with Gauss's principle, of such 

 exceptional utility in the development of modern physics; finally 

 the development of the Leibnitzian doctrine of work and vis viva 

 into the law of the conservation of energy, which more than any 

 other principle has consciously pervaded the progress of the nine- 

 teenth century. Clausius's theorem of the Virial (1870) and Jacobi's 

 (1866) contributions should be added among others. 



The potential, though contained explicitly in the writings of 

 Lagrange (1777), may well be claimed by the last century. The 

 differential equation underlying the doctrine had already been 

 given by Laplace in 1782, but it was subsequently to be completed 

 by Poisson (1827). Gauss (1813, 1839) contributed his invaluable 

 theorems relative to the surface integrals and force flux, and Stokes 

 (1854) his equally important relation of the line and the surface 

 integral. Legendre (published 1785) and Laplace (1782) were the 

 first to apply spherical harmonics in expansions. The detailed devel- 

 opment of volume surface and line potential has enlisted many 

 of the ablest writers, among whom Chasles (1837, 1839, 1842), 

 Helmholtz (1853), C. Neumann (1877, 1880),Lejeune-Dirichlet (1876), 

 Murphy (1833), and others are prominent. 



The gradual growth of the doctrine of the potential would have 

 been accelerated, had not science to its own loss overlooked the 

 famous essay of Green (1828), in which many of the important 

 theorems were anticipated, and of which Green's theorem and 

 Green's function are to-day familiar reminders. 



Recent dynamists incline to the uses of the methods of modern 

 geometry and to the vector calculus with continually increasing 



