G32 PROCEEDINGS OF THE AMERICAN ACADEMY. 



proceeding is not always necessary is that cycles of the first order very 

 often may be thought of simply as those special cases of second order 

 cycles which happen to lie on certain curves in the P Q plane. Thus 

 a complete representation of the properties of the cycle of Figure 1 will 

 turn up, quite incidentally, in the last part of this paper. 



Perhaps the best way to bring out the scope and power of the graphical 

 method proposed will be to carry through, in some considerable detail, 

 an application of it to a definite problem ; an excellent problem for this 

 purpose is presented by the cycles which come up in connection with 

 such turbines as use air instead of steam as the working substance. 

 The simplest case would be that of a turbine of the De Laval type, 

 in which the working substance expands adiabatically from a high 

 pressure to the pressure of the exhaust in a single nozzle so shaped 

 that the available energy in the working substance appears as kinetic 

 energy. The momentum of the moving stream is then used to turn 

 a wheel. The sequence of processes in such a machine* is as follows : 



1. Air is taken in at atmospheric pressure and temperature (corre- 

 sponding to the point a of Figure 2) and compressed. 



2. It is then heated; this will be done at constant pressure, for the 

 pressure cannot increase if a continuous flow is to be kept up, and if it 

 diminishes during the heating, energy is needlessly lost. (This process 

 is represented by the horizontal line ending at .rf.) 



3. There is adiabatic expansion in a nozzle (line de). 



4. Either the hot exhaust (point e) is thrown away at atmospheric 

 pressure and new air is taken in, or the same working substance is 

 cooled and used over again. (Either process is represented by the 

 line ea.) 



The only thing not yet described is the nature of the compression, 

 and this may be adiabatic (line ac) or isothermal (line ab) or anything 

 between. The two simplest cases may be taken as typical ; the corre- 

 sponding cycles will be called 



Type A. TJie Adiabatic or Brnyton Cycle (a c d e a of Figure 2). 



Type B. The Isothermal Cycle (ab d e a of Figure 2). 



Of tlie non-atmospheric cycles omitted from the list on p. 417 as being not "ac- 

 curately defined," all but cycle V (of p. 233) are of the third order; such cycles 

 would require three variables and a three-dimensional graphics, which would bear 

 to the P Q plane exactly the same relation which that plane bears to the Q axis 

 of Dr. Lucke's paper. Luckily very few important cycles are of higher order 

 than the second. 



* Here, as elsewhere in this paper, only purely ideal conditions are considered. 



