148 THE EEV. S. EAENSHAW OX THE SIATHE^IATICAL THEOET OF SOUXH. 
applicable to the problem of sound at all. Many analytically approximate forms might 
be invented and used for Boyle’s law, and each would have its peculiar physical attri- 
butive effects on the sound-wave; and we might thus, by adopting first one and then 
another of these analytically approximate laws, invent ad libitum an inexhaustible hst 
of properties of the sound-wave which have no real existence where Boyle’s law is 
strictly true. From which therefore it would seem to be a necessary consequence, that 
an equation between^ and must not only be analytically but also physically approxi- 
mative, in order that the results deduced from it may be accepted as real approximations 
to the true laws of nature. 
dfiL dc 
45. By means of the expressions for ^ and we may not only discover the proper- 
ties of motion in a tube without having recourse to the usual equations, when the rela- 
tion between and p is known, but we may also solve many inverse problems. 
Also, if the tube be filled with a medium of such a nature that the relation between 
and ^ changes continuously from point to point, or is difierent in different parts, yet 
if 
has the same value everywhere, waves will travel through the tube with a 
uniform velocity. 
If the nature of the medium should vary slowly and continuously, the velocity of the 
wave-transmission would be known, from the equations given above, by integration. 
46. If, through the partial radiation of heat, or from any other cause, the d^Tiamicai 
equation should take the form 
df' J \dx' 
we must integrate it as before by assuming 
dt J dx^' 
which gives 
(Fa)^=/(«, Fee). 
This equation being integrated -will furnish the form of 
proposed dynamical equation will be 
F; 
and then the integral of the 
-j- (C+ Fa)if -f- (pa, 
[a?=+Fa.#— (p'a, 
which does not present any new difficulty. 
