630 PROCEEDINGS OF THE AMERICAN ACADEMY. 



tions in which they occur, and to assume for them constant " standard" 

 values. The dependence of the results upon these values will then be a 

 separate, and usually a very simple problem. For the determination of 

 the dimensions of" a cycle two more coordinates are usually needed, 

 which may be thought of as describing in some generalized way the 

 height and breadth of the cycle. Thus for a Carnof cycle the tempera- 

 ture and entropy ranges might be used, or the pressure or volume ranges 

 along two adjacent sides, or some less obvious coordinates, as, for in- 

 stance, the quantities of heat involved in two successive transformations. 

 It is hoped that this paper will show that in a very great number of 

 cases it is advantageous to choose as these two coordinates the ratio of 

 compression (P^/Pa), which will be denoted by P, and the heat taken 

 in from outside sources by a unit mass of working substance during its 

 passage once around the cycle, which will be denoted by Q. The 

 chief reasons for this choice are to be found in the simplicity of the 

 formulae to which it leads, and in the ease with which the physical inter- 

 pretation of these formulae can be brought out. It may also be noticed 

 that these quantities are simply and independently controllable and are 

 easily measured. 



If values of Q be taken as abscissae and values of P as ordinates, the 

 result is 3l P Q plane upon which a point represents and completely de- 

 scribes a thermodynamic cycle (when its type and the values of the 

 parameters are known) in exactly the same way that a point on the p v 

 plane represents and completely describes a thermodynamic state (when 

 the form of the characteristic equation is known, together with the values 

 of the constants which it involves). Any property {X) of such a cycle 

 can then be plotted along a third rectangular axis, and the resulting three- 

 dimensional surface will give a complete picture of that property for all 

 cycles of the type under consideration. If such a surface be constructed 

 for each different cyclic type to be studied, one can read off from this 

 set of models any desired information whatever about the property to 

 which they correspond. The simplest way to handle such surfaces is to 

 map their contour lines (A^ =: a constant) on the basal PQ plane, just 

 exactly as isothermals or adiabatics are mapped on the familiar p v plane 

 of thermodynamic states. 



We have said that two coordinates will usually be needed to deter- 

 mine the dimensions of a cycle, but in many exceptional cases one is 

 enough. An example is a cycle which consists of an isothermal, an 

 isopiestic, and an adiabatic (Figure 1). The point a, when it is fixed by 

 the parameters, determines the isothermal and the adiabatic upon which 



