PROFESSOR H. B. DIXON OX THE RATE OF EXPLOSION IN GASES. 
153 
is only possible for a particular relation between tbe pressure and density of the gas, wbich is different 
from tbe one actually bolding. In tbe case of tbe explosion-waves it seems possible, bowever, that tbe 
temperature, pressure, and density of tbe gas should so adjust themselves as to make Riemann’s equations 
applicable. In fact, they must do so if the front of tbe wave keeps its type, wbich it probably does 
when tbe velocity has become constant. 
If pQ and p are tbe densities of tbe gas in the undisturbed state, and at some point in tbe wave 
respectively, then, refeiu-ing tbe motion to a system of coordinates moving forward with the velocity V 
of the wave, we have tbe condition of steady motion 
pu = p^Y 
where u is the velocity at tbe point at which tbe density is p ; 
also 
dp du 
/ = - • y- • 
ax cix 
As pu is a constant quantity we may integrate tbe latter equation and obtain 
p - Po= - pu (m - Uq) = Y~ (p - Pq) , 
P 
which gives for tbe velocity of tbe wave propagation 
V = 
(P - Pa) _ P 
ip — do) do 
This equation gives a relation between the density p at any point of the Avave and tbe pressure p at 
that point, tbe pressure and density in the initial state being and p^. Tbe equation can be simplified 
if p is large compared to />q for 
d ^ 1 / (1 
do (d — do) do / 
do//') 
so that we may write 
1 _ 
— do/d 
Putting, for the sake of argument, p = 10/>q, the second square root in the aboA^e expression becomes 
1-05, so that as a first approximation it may be put equal to one. Writing P for the excess of pressure 
over the atmospheric pressure, the equation noAv takes tbe simple form 
In this expression the value of P has to be Avritten in dynes per square centimetre, and tbe velocity 
would be given in centimetres. For more convenient numerical calculation Ave may adopt provisionally 
tbe suggestion of Guillaume that a pressure of 75 cms. of mercury under tbe normal conditions should 
be called an atmosphere. Tbe atmosphere then Avould be exactly one megadyne per square centimetre, 
and if P is given in atmospheres, the equation becomes 
or if \ is expressed in metres per second. 
MDCCCXCIIT.—A. ~ X 
