THE EEV. S. EAENSHAW ON THE MATHEMATICAL THEOEY OF SOUND. 147 
42. And if c denote the velocity of transmission of the density then we have 
consequently 
dc 1 d / ^dp\^ 
dq dq\^ dq) 
' Now the former of these equations shows that unless the term 
be constant, the 
property of the superposition of wave-transmission on particle-velocity, proved in art. IG, 
does not hold good. But if it be constant, then -{-(«/'; which is the general relation 
between pressure and density when that principle of superposition holds good. Hence, 
as mentioned in art. 36, the development of heat puts an end to this property in all 
known gases. 
43. In the case of negative waves we may institute a method of reasoning similar to 
that employed in arts. 23 et seq., and arrive at analogous results. We shall also find 
that, taking c=(^^ —u for this case, the maximum value of SfW will occur in that 
section of the tube where c=0 ; from which it follows that at that section 
which is always possible and finite. Hence may be determined the limit to the quantity 
of a gas that can pass through a pipe in a given time, even into a vacuum. 
44. The expression in art. 42 for ^ 
shows that c is in general a function of so that 
in general there will be a constant change of type. In one case, however, there will be 
no change of type. This wall take place when ^=0, that is when is constant. 
Assume for this case 
This equation expresses the nature of the medium which is distinguished by the pro- 
perty, that it transmits waves without change of type. And if we pass from this to the 
dynamical equation, we find 
d'^tj B d’^y 
dt’^ q^ dx^ 
Now it has been usual to reduce the equation (3.) to this form for the purposes of 
approximation ; but the process appears to be allowable only so far as the equation 
i^=A— — may be taken to be a physical approximation to Boyle’s law To me it 
does not seem to be an allowable approximation ; and consequently I do not consider 
the solution of the dynamical equation, which has been obtained by this means, to be 
