252 Mr. S. H. Burburv on 



determine that, a physical assumption has to be made 

 with regard to the motion, by virtue o£ which assumption 

 the algebraic theorem becomes applicable. This is done by 

 both Boltzmann and Planck. Archimedes proposing to 

 move the earth begins by assuming a locus standi. 



2. In the present paper I propose to consider certain diffi- 

 culties which to me at least present themselves in the late 

 Professor Willard Gibbs' ' Principles of Statistical Mechanics,' 

 Chapter XII. In this chapter he seeks to establish a general 

 law for the variation of entropy in ensembles of systems, 

 without, as I understand him, making any special physical 

 assumptions, by the sim])]e use of the typical algebraic 

 theorem. To explain my difficulties it is necessary to set out 

 the method developed in Chapter I. of the same work ; and 

 this is fortunately easv, owing to the extreme simplicity and 

 elegance in which he has himself presented it. 



3. A material system is defined by the n generalized co- 

 ordinates qi.. .qn, and. corresponding momenta jJi. . .pn- Each 

 system is at all times free from the influence of any bodies 

 external to itself, including all other systems, but with the 

 exception of forces to fixed centres, such as gravitation. Its 

 whole energy is therefore constant for all time. The '' co- 

 ordinates," however, including here in that term both ^'s 

 and q's, vary. The system passes through a series of phases 

 defined by the values of qi...p„. All phases in which 

 the *• coordinates ^' lie between the limits q^...qi-\-dqi... 

 Pn-.-pn + dpn coustitute tlio e.vtension in phase dqi...dpn. 

 And by Liouville^s theorem dqi...dpn is constant for the same 

 group of systems. 



4. Xow suppose a very great number of such mutually 

 independent systems. There will at a given instant be many 

 systems within a given extension m phase. We are now to 

 consider the extension in phase which lies between jo/ and 

 p\'...qn and qn" . The difierences p\ —p\ &c. are in- 

 finitesimal ; and he assumes systems to be distributed 

 continuously between these limits, a state of things which, 

 as he points out, requires us to suppose the number of systems 

 indefinitely great. In fact, since no system interferes with 

 any other system, there is no limit to the number which 

 may be in exactly the same phase. 



5. The number of systems between the limits p^ —p\ &c. 

 is now denoted by the continued product 



D(i'i"-p/)(yV'-/V) ■•••(?."-?»'), 



in which D is a function of pi.-..qn, and is called the density 

 in pliase aX p^ — qn. 



