452 Mr. A. W. Rucker on the [June 18, 



vent se couper," and offers a proof which rests upon the assumption 

 that if a body could undergo a series of operations represented as to the 

 changes of pressure and volume by PQMP (where PQ is an isothermal 

 and PM, QM two adiabatics), no heat would be gained or lost at any part 

 of the cycle except PQ. 



It is, however, evidently impossible that the body could, at the point 

 M, pass from one adiabatic to another without absorbing or emitting heat, 

 t. e. while fulfilling the very condition that it should not pass from one 

 adiabatic to another ; and the question as to the possibility of the inter- 

 section of two adiabatics must therefore be submitted to a more general 

 investigation, as it is certainly conceivable that heat might be gained or 

 lost during the passage from the point M considered as lying on the first 

 curve to the point M considered as belonging to the second, whether it 

 took place, as supposed by M. Verdet, without any accompanying changes 

 of pressure or volume, or whether, as we shall see would be generally the 

 case, it could only be accomplished if the body were caused to assume a 

 series of intermediate states involving such changes. 



The question admits of an easy answer if we consider the case of 

 bodies which can exist in two distinct states under the same circumstances 

 of pressure and volume ; and for the present we may confine our atten- 

 tion to water, which is the most conspicuous representative of the class, 

 and which, at the ordinary atmospheric pressure and at temperatures 

 between C. and 4 C., exists in a series of states in which the volumes 

 are the same as those which it assumes if heated at the same pressure 

 from 4 C. to about 8 C. 



Hence whereas for higher temperatures all the properties of water 

 at atmospheric pressure are completely defined if we know the volume, 

 such is not the case between the limits above indicated ; but each point 

 on the line of constant pressure given by p=l atmosphere between its 

 intersections with the isothermals C. and 4 C. corresponds to two 

 states of the water, or rather, since if the water-substance be converted 

 into ice it will, if cooled sufficiently, again pass through the same range 

 of volumes, each point corresponds to three states and is the intersection 

 of three isothermals ; and as a similar remark may be made with respect 

 to neighbouring lines of constant pressure, it follows that there is a 

 region in the plane of pv such that three states of the water-substance 

 correspond to % each point within it, and that therefore the values of p 

 and v given by any such point do not define the state of the water. 



If, however, from every point in the plane of pv we draw perpendicu- 

 lars to that plane, proportional to those values of some other property of 

 the water (say, in this case, its temperature) which correspond to the 

 conditions of pressure and volume represented by the points from which 

 they are drawn, the extremities of such ordinates will form a surface 

 which will be met once, or more than once, by any particular ordinate, 

 according as the water can exist under the circumstances of pressure 



