66 Trans. Acad. Sci. of St. Louis. 



by a moving point on the thermodynamic surface whose 



equation is 



PV = CT. 



When a gas is heated either under constant pressure or 

 constant volume, the moving point on this surface traces a 

 straight line in space. There are, of course, an infinite 

 variety of operations to which the gas may be subjected, in 

 which the point representing the condition of the gas may 

 trace out in each case some definite path on the surface re- 

 ferred to. Each operation will involve some value for the 

 specific heat. 



The average density of the mass M is at all times three 

 times the density at its surface. Hence calling F the initial, 

 and Fthe final volume of the spherical mass, the law of gases 

 gives the equations 



P V = ICT M (9) 



PV=r d OTM. (10) 



These equations may also be obtained from (2) and (5) by 

 multiplying by the volumes of the respective spheres having 

 radii B and R. The right hand member is then reduced to 

 the form given, by introducing the value of M or the equal 

 value Mfvom (3), or the equation which follows (5). 



Since 



P T 4 

 p- ip = Q , 



the value of P in the last equation may be eliminated. The 

 two equations then give 



T*V=T zV =*TrB *T> 



which by (3) becomes 



•ns "it — _ 



6 V C J 



7 T 3 F= _(__). (11) 



The point traces upon the surface a path, which projects 



