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PROFESSOR W. J. M. RANKINE ON THE THERMODYNAMIC 
as non-conducting, because of there being an indefinite length of matter before the fore- 
most and behind the aftermost plane, to resist conduction. 
The transfer of heat, therefore, takes place wholly amongst the particles undergoing 
variation of pressure and bulkiness ; and therefore for any given particle, during its 
passage from the front to the back of the wave, the integral amount of heat received must 
he nothing; and this is the thermodynamic condition which gives the required equation. 
That equation is expressed as follows : 
in which r denotes absolute temperature, and <p the “ thermodynamic function.” The 
value of that function, as explained in various papers and treatises on thermodynamics, 
is given by the following formula : 
<p = Jc hyp lo gr+xW+f, (12 a) 
in which J is the dynamical value of a unit of heat ; c, the real specific heat of the 
substance; %(r), a function of the temperature alone, which is =0 for all temperatures 
at which the substance is capable cf approximating indefinitely to the perfectly gaseous 
state, and is introduced into the formula solely to provide for the possible existence of 
substances which at some temperatures are incapable of approximating to the perfectly 
gaseous state ; and U, the work which the elastic forces in unity of mass are capable 
of doing at the constant temperature r. The substitution for the integral in equation 
(12) of its value in terms of p and s for any particular substance, gives a relation between 
the limits of pressure^), and p 2 , and the limits of bulkiness s x and s 2 , which being com- 
bined with equation (5), or with any one of the equivalent equations (6), (8), or (9), com- 
pletes the expression of the laws of the propagation of waves of finite longitudinal dis- 
turbance and permanent type in that particular substance. 
§ 7. Assumption as to Transfer of Ileat. — In applying the principles of the preceding 
section to the propagation of waves of longitudinal disturbance, it is obviously assumed 
that the transfer of heat takes place between the various particles which are undergoing 
disturbance at a given time, in such a manner as to ensure the fulfilment of the dyna- 
mical condition of permanency of type. It appears highly probable, that how great 
soever the resistance of the substance to the conduction of heat may be, that assumption 
as to the transfer is realized when the disturbance is sudden , as described in § 5 ; for 
then particles in all the successive stages of the change of pressure and bulkiness within 
the limits of the disturbance are at inappreciable distances from each other ; so that the 
resistance to the transfer of heat between them is inappreciable. 
But when the disturbance is not sudden, it is probable that the assumption as to the 
transfer of heat is fulfilled in an approximate manner only ; and if such is the case, it 
follows that the only longitudinal disturbance which can he propagated with absolute per- 
manence of type is a sudden disturbance. 
§ 8. Combination of the Dynamic and Thermodynamic Equations. — In every fluid, and 
