Dr. Rankine on the Acceleration and Retardation of Streams. 289 



place where the pressure is p , and the potential energy of attrac- 

 tion U , and finally arrives at a place where the quantities p and 

 U have their original values p ] and Uj. Let the fluid be in the 

 condition called adiabatic — that is, let it neither receive nor give 

 out heat. Then the relation between p and s is defined by the 

 constancy of the thermodynamic function 



£=Jchyp.logT+x( T )+ X-^dsi 



P) 



in which J is the dynamical equivalent of a unit of heat, c the 

 real specific heat of the fluid, t the absolute temperature, %(t) a 

 function of t which is null for substances capable of approxima- 

 ting indefinitely to the perfectly gaseous state, and will be 

 omitted throughout the rest of this paper ; and in the integral 



dp 



— is taken on the supposition that s is constant. Then at the 



place where the pressure is p 0) the energy of flow in each unit of 

 mass is expressed by 





sdp, .... (3) 



'Pa 



the integral being taken subject to the condition that the ther- 

 modynamic function </> has a certain constant value. 



By the time that the stream has arrived at the place where U 

 and/> return to their original values, v also has returned to its 

 original value v l ; and here there is no thermodynamic accelera- 

 tion or retardation. 



But next suppose that at the place where the pressure is p 

 each unit of mass has a certain quantity of heat either added to 

 or abstracted from it, so as to change the thermodynamic func- 

 tion from cf> to cj> ! . That quantity of heat is expressed in dyna- 

 mical units by 



P ' M (4) 



j. 



Let the return to the original pressure take place with this altered 

 value of the thermodynamic function. Then throughout this 

 second division of the stream each value p of the pressure will 

 have corresponding to it a value s 1 of the bulk in ess suited to the 

 new value of the thermodynamic function, and different from the 

 value s corresponding to the same pressure in the first division 

 of the stream. The relation between the change in the value of 

 <f> and the change in the values of s is given by the equation 



t C s dp 

 0-£'=Jchyp.log- f + !^«fe (5) 



