A. MATHEMATICS AND ASTRONOMY. 5 



pulsion of the points of the system, all the integral equations 

 of the motions can be represented by the partial differential 

 quotients of a certain function, called the Primary Function, 

 of their co-ordinates in a manner similar to that in which, ac- 

 cording to La Grange, the differential equations of the mo- 

 tions can be represented by the partial differential quotients 

 of a function known as La Grange's function of the forces. 

 The primary function of Sir William Hamilton is a complete 

 solution of the partial differential equations of La Grange's 

 function, as was shown by Jacobi. The integration of this 

 differential equation was developed by Jacobi, since whose 

 time the theory has undergone expansion in two respects, 

 by Zipschitz and Sobering, to whose researches Mtiller adds 

 the following propositions : First, the sum of such changes in 

 the primary function and in the expenditure of force as may 

 be produced by the variations of the initial and final co-ordi- 

 nates alone, is, in the variation of every motion that presup- 

 poses a force function, and neither explicitly nor implicitly 

 contains the time, equal to zero. This proposition he desig- 

 nates as "The principle of Energy." Correlated to the pre- 

 ceding is Miiller's second proposition, which he calls "The 

 principle of Action," which may be enunciated as follows : 

 That change of the action which is conditioned by the varia- 

 tion of the initial and final co-ordinates alone vanishes with 

 the change of every motion that presupposes a force function, 

 and does not contain the time either explicitly or implicitly. 

 Here, as in the previous proposition, if we imagine the whole 

 series of constantly altered motions to be run through with, 

 they will in general be distinguished by different values of 

 potential and kinetic force and energy ; in proportion as by 

 the mere alteration of the co-ordinates the potential dimin- 

 ishes, so does the kinetic increase. These propositions, which 

 are represented by Mtiller in algebraic language, are exem- 

 plified by several applications. Applying the first proposi- 

 tion to a simple case, he by it develops the motion of the or- 

 dinary pendulum ; but his most interesting results relate to 

 the theory of heat. If according to the mechanical theory 

 heat be considered as molecular motion, the application to 

 this hypothesis of Miiller's "Principle of Energy" leads im- 

 mediately to the well-known first law of thermo-dynamics; 

 while, if we apply to these molecular motions the theorem of 



