324 Dr. C. Burton on Plane and Spherical 



Since we assume that there is no transference of heat by 

 conduction or radiation, the rave at which the total energy of 

 the system increases must be equal to the rate at whicb work 

 is being done upon it by the piston A. Let 6 be the abso- 

 lute temperature to the right of B, that between A and B 

 being 6', and let us further assume for simplicity that 



±- = a const. ; 

 p6 



while 7, the ratio of the two specific heats, is also supposed 

 constant. It can then be shown without difficulty that the 

 total energy per unit mass between A and B exceeds that to 

 the right of B by 



( 7 -l)po 0o 2 ' 



and multiplying this by pqV', the mass of air which crosses 

 one unit of the surface B in each unit of time, we obtain the 

 rate (referred to unit area) at which the system is gaining 

 energy. Again, the rate at whicb unit area of the piston does 

 work on the system 



p'e' 



and equating this to the rate of gain of energy, we obtain 



*&-* V i + G^ ( *-* ) - • • (9) 



We may also write equation (8) in the form 



p < v (V<-v)=^(p<0-pA); ■ ■ ■ (10) 



and (7), (9), and (10) will then serve to determine V, /o f , & 

 when v, p , 6 are given. Since we have taken all these 

 quantities to remain constant throughout the motion, we see, 

 as before, that at each instant all the necessary conditions are 

 satisfied ; the principles of mass and momentum, together 

 with our supposition that there is no exchange of heat, being 

 sufficient to determine what takes place at B. Again, if at a 

 time t from the commencement of the motion we take the 

 distance of B from A to be (V' — v)t, so that initially B coin- 

 cides with A, the initial conditions are satisfied. The assumed 

 motion thus satisfies all the necessary conditions, and is there- 

 fore the actual motion. 



8. If we compare the results of the last two sections with 



