286 
PROFESSOR W. J. M. RANKINE ON THE THERMODYNAMIC 
in which A and B are arbitrary constants. The value of A depends on the magnitude 
of the disturbance, and that of B upon the position of the point from which x is reckoned. 
In applying these general equations to particular substances, the values of r and <p are to 
be expressed in terms of the pressure j), by the aid of the formulae of the preceding 
section; when equation (33 b) will give the value of# in terms of p, and thus will show 
the type of disturbance required. 
Our knowledge of the laws of the conduction of heat is not yet sufficient to enable 
us to solve such problems as these for actual substances with certainty. As a hypo- 
thetical example, however, of a simple kind, we may suppose the substance to be per- 
fectly gaseous and of constant conductivity. The assumption of the perfectly gaseous 
condition gives, according to the formulae of the preceding sections, 
PS (7 + t)(Pi +Pz)p~ 2 P z 
(7-l)Jc f ’ (7 + l)(Pi+P 2 ) p -2P 2 ’ 
and 
It is unnecessary to occupy space by giving the whole details of the calculation ; and 
it may be sufficient to state that the following are the results. Let 
P- 
Pi +Pi —„ 
2 — 
2 ’ 
then 
dx dx 
dp dq (y -\-l)mJc s ’ 
k 
(y— l)(Pi+^ 2 )~ 4 g 
q\-q l 
X=- 
f (7 — ViP i +Pz) * hyp i 0 g _]_ 2 hyp i 0 g (\_L 
1 2 q } b ?i -q^ 3if 6 V q\ 
(34) 
(7+l)mJc s ^ Q- n - 1 & o, — a 1 P a 2 1 (' ' (^A) 
In equation (34 a) it is obvious that x is reckoned from the point where q— 0; that 
is, where the pressure p = PjL ~ l ; a mean between the greatest and least pressures. The 
direction in which x is positive may be either the same with or contrary to that of the 
advance of the wave ; the former case represents the type of a wave of rarefaction, the 
latter that of a wave of compression. For the two limiting pressures when q—±q„ ^ 
becomes infinite, and x becomes positively or negatively infinite ; so that the wave is 
infinitely long. The only exception to this is the limiting case, when the conductivity Tc 
is indefinitely small ; and then we have the following results: when p=p x , or p=p 2 , 
dx 
is infinite, and x is indefinite ; and for all values of p between p x and p 2 , ^ and x are 
each indefinitely small. These conditions evidently represent the case of a wave of 
abrupt rarefaction or compression, already referred to in §§ 6 and 7. 
