of the Mechanical Theory of Heat with Hamilton's Principle. 369 

 ration of c. We can therefore in this case write instead of (3 a) : — 



SU-^Sc=|8t?'+mi;^logz, . . . [U) 



by which the characteristic peculiarity of the quantity which 

 stands on the left-hand side of my equation comes out still more 

 distinctly. 



In order that we may conveniently compare together the three 

 mutually related equations with which we have to do in the 

 newer considerations on the mechanical theory of heat, in refer- 

 ence to their applicability, it will probably be advisable to sum- 

 marily recapitulate the differences which prevail between them. 

 The equation which expresses the principle of least action in its 

 original form, and which in our notation is 



a(w^2)=0, (4) 



presupposes that the ergal is represented by an invariable func- 

 tion of the coordinates, and also that the energy has an invari- 

 able value. In the Hamilton's equation (2a) an invariable func- 

 tion is likewise presupposed, but the energy may vary. Lastly, 

 in my equation both an alteration of the energy and also of the 

 function representing the ergal are admissible. 



The latter generalization was absolutely necessary for the ap- 

 plication to the theory of heat, because, with the changes of state 

 of a body which there come into consideration, occur variations 

 of the effective forces which are independent of the space-coor- 

 dinates and cannot be represented by an ergal of invariable form. 



Besides this, other difficulties are met with in the cases to be 

 considered in the science of heat, which render the immediate 

 appplication of the Hamilton's equation (2) inadmissible. 



Equation (2) presupposes that all the material points in the 

 system under consideration take for their motion one common 

 time i, which with the change of motion changes in like manner 

 for all the points. But if we conceive a body as a system of a 

 great many moving material points, and make even the simplified 

 assumption that all the points move in closed paths, yet we are 

 not at liberty to presuppose that they all describe their paths in 

 the same time^ and that with a change of state of the body all 

 the times of revolution change in like manner. Consequently, 

 to take into account this circumstance, special considerations are 

 necessary. 



The difficulties become still greater when we drop the assump- 

 tion that all the points move in closed paths, and admit that their 

 motions are quite irregular. In M. Szily's analysis, however, 

 not one of these difficulties is mentioned. 



Bonn, May 1872. 

 Phil, Mag. S. 4. Vol. 44. No. 29i. Nov, 1872. 2 B 



