276 Prof. J. J. Miiller on a Mechanical Principle 



stance, satisfy Weber's law, or motions in Gauss's and Riemann's 

 space of multiple dimensions. 



The slight improvement of the physical side of Hamilton's 

 method stands far from the high degree at which these analytical 

 investigations have arrived. An essential peculiarity of it con- 

 sists in this — that it passes from a given motion of the system 

 of points to another in a similar manner to that in which La- 

 grange's process passes from one configuration of the points to 

 another. The primary function, a definite integral which is ex- 

 tended over the original motion, undergoes an alteration by the 

 variation of the arbitrary constants of the motion ; and this va- 

 riation, or that of a similar integral representing the expenditure 

 of the force, is given by Hamilton's symbolic equations of mo- 

 tion. Hence Hamilton's method differs, secondly, from La- 

 grange's (in which the force-function changes according to the 

 elements of the given motion) in the same way as the variation 

 differs from the differentiation of the functions. This second 

 aspect of it could not but lead immediately to a new treatment 

 of the perturbations, which has by Hamilton, Jacobi, and Sche- 

 ring been developed into a series of new systems of perturbation- 

 formula?. Only the above-mentioned application of the variation 

 of the motion, which is in principle only a particular way of re- 

 presenting the latter, is not the essential of the new view ; that 

 must much rather be sought in similar principles to those on 

 which the ordinary differential equations of the mechanical pro- 

 blem are based. It is true that one signification of these prin- 

 ciples, the representation of individual integrals, does not here 

 come into consideration in the indicated general process of inte- 

 gration; their physical meaning, however (independent of the 

 other), which was proved most evidently in the proposition of 

 the vis viva, especially with the generality given to it by Helm- 

 holtz, remains here also; and this justifies an examination of it. 

 Such an examination of the physical aspect of Hamilton's 

 method is attempted in the sequel. It appeared the more re- 

 quired, as the endeavours of physicists to deduce the second 

 proposition of the mechanical theory of heat in a similar manner 

 as the first, from purely mechanical conceptions, clearly per- 

 mitted the supposition of a new mechanical principle. Boltz- 

 mann*, Clausiusf, and Ledieu % have succeeded in obtaining 

 from Lagrange's differential equations the proposition mentioned : 

 it did not, however, like the first proposition, come from a uni- 

 versal principle ; but, on the contrary, those investigations led 

 to new mechanical propositions, which certainly did not possess 



* Wiener Sitzungsberichte, vol. liii.; Pogg. Ann. vol. cxliii. p. 211. 

 t Pogg. Ann. vol. cxlii. p. 433. Phil. Mag. S. 4, vol. xlii. p. 161. 

 X Comptes Rendus, 18/3, 1874. 



