278 
PROFESSOR W. J. M. RANKIN E ON THE THERMODYNAMIC 
time, being each, in unity of time, expressed by the mass-velocity ; hence we have, as 
the cinematical condition of uniformity of type, the following equation : 
a — u. a — Uq a 
— — = -=o=m. . . 
Si s 2 b 
Another way of expressing the same condition is as follows : 
A u— — mAs 
( 2 ) 
( 3 ) 
§ 4. Dynamical Condition of Permanency of Type. — Let p x and p 2 be the intensities 
of the longitudinal pressure at the foremost and aftermost advancing planes respectively. 
Then in each unit of time the difference of pressure, p. 2 — p t , impresses on the mass m 
the acceleration u 2 —u 19 and consequently, by the second law of motion, we have the 
following value for the difference of pressure : 
p 2 -Pi=m(u. 2 — u x ) (4) 
Then substituting for the acceleration u 2 —u 1 its value in terms of the change of bulki- 
ness as given by equation (3), we obtain, for the dynamical condition of permanency of 
type, the following equation, 
P2—Ti=tn\s l —s 2 ), (5) 
which may also be put in the form of an expression giving the value of the square of 
the mass-velocity, viz. 
W 2___A p__dp 
111 As ds 
( 6 ) 
The square of the linear velocity of advance is given by the following equation : 
a 2 =m 2 S 2 =-S 2 f (7) 
ds v 
The integral form of the preceding equations may be expressed as follows. Let S, as 
before, be the bulkiness in the undisturbed state, and P the longitudinal pressure ; then 
in a wave of disturbance of permanent type, we must have the following condition ful- 
filled : 
( 8 ) 
§ 5. Waves of Sudden Disturbance. — The condition expressed by the equations of the 
preceding section holds for any type of disturbance, continuous or discontinuous, gradual 
or abrupt. To represent, in particular, the case of a single abrupt disturbance, we must 
conceive the foremost and aftermost advancing planes already mentioned to coalesce into 
one. Then P is the longitudinal pressure, and S the bulkiness, in front of the advancing 
plane ; p is the lonigtudinal pressure, and s the bulkiness, behind the advancing plane ; 
and the advancing plane is a wave-front of sudden compression or of sudden rarefaction * 
* (Note, added lsi August, 1870.) Sir William Thomson has pointed out to the author, that a wave of 
sudden rarefaction, though mathematically possible, is an unstable condition of motion ; any deviation from 
absolute suddenness tending to make the disturbance become more and more gradual. Hence the only wave 
of sudden disturbance whose permanency of typo is physically possible, is one of sudden compression ; and this 
is to be taken into account in connexion with all that is stated in the paper respecting such waves. 
