1874.] Adiabatics and Isothermals of Water. 459 



i. e. according as there is or is not enough steam to supply by its conden- 

 sation a sufficient quantity of heat to melt all the ice ; and as 



which is always positive, as s o- is negative, we have the largest and 

 smallest volumes given by the limits 



#=0 and 1 # =0, 

 or #=0 and =0. 



The maximum volume is therefore in any case given by 



and the minimum volume is 



(a-,) r (l-*o)-pg + y in the first 

 r-p 



and (s a) rtr +p + a in the second case ; 



r 



and the differences between these quantities give the range of volumes 

 for which the adiabatic belonging to the initial values a? , coincides with 

 the isothermal. 



In conclusion it is only necessary to point out that some of the results 

 in the earlier part of the paper follow immediately from the ordinary 

 formulae of thermodynamics. 



If Cp and Op are the specific heats at constant pressure and constant 

 volume respectively, and if, to avoid confusion, we write the quantity 

 which is supposed to remain constant as a subscript to a partial differ- 

 ential coefficient, we have the well-known expressions 



p 

 and l^L\ =^^ 



where Q is constant for any adiabatic. From the first it follows that 



*"(!) => 



and 



\pq \pt 



i. e. the adiabatics and isothermals touch one another at points of maxi- 

 mum or minimum volume. 

 Also by differentiation, 



C, dV d C\ dv 



