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VI. On the Geometrical Representation of the Expansive Action of Heat, and the 
Theory of Thermo-dynamic Engines. By William John Macquorn Rankine, 
C.E., F.R.SS. Lond. and Edin. 8fc. 
Received December 5, 1853, — Read January 19, 1854. 
Section I.— INTRODUCTION AND GENERAL THEOREMS. 
(Article 1.) The first application of a geometrical diagram to represent the ex- 
pansive action of Heat was made by James Watt, when he contrived the well-known 
Steam-Engine Indicator, subsequently altered and improved by others in various 
ways. As the diagram described by Watt’s Indicator is the type of all diagrams 
representing the expansive action of heat, its general nature is exhibited in fig. 1. 
Let abscissae, measured along, or parallel to, the axis 
OX represent the volumes successively assumed by a given 
mass of an elastic substance, by whose alternate expansion 
and contraction heat is made to produce motive power ; 
OV A and OV B being the least and greatest volumes which 
the substance is made to assume, and OV any intermediate 
volume. For brevity’s sake, these quantities will be de- 
noted by V A , V B , and V, respectively. Then V B — V A may 
represent the space traversed by the piston of an engine 
during a single stroke. 
Let ordinates, measured parallel to the axis OY and at 
right angles to OX, denote the expansive pressures successively exerted by the sub- 
stance at the volumes denoted by the abscissae. During the increase of volume from 
V A to V B , the pressure, in order that motive power may be produced, must be, on 
the whole, greater than during the diminution of volume from V B to V A ; so that, 
for instance, the ordinates VP, and VP 2 , or the symbols P, and P 2 , may represent 
the pressures corresponding to a given volume V during the expansion and con- 
traction of the substance respectively. 
Then the area of the curvilinear figure, or Indicator-diagram, AP,BP 2 A, will repre- 
sent the motive power, or <£ Potential Energy,” developed or given out during a com- 
plete stroke, or cycle of changes of volume of the elastic substance. The algebraical 
expression for this area is 
Q 2 
Fig. 1. 
Y 
(I*) 
