ON LAGRANGE’S BALLISTIC PROBLEM. 
171 
the position specified by x 0 . At the time t = 0 the gas is supposed to be at rest. Let 
p 0 and p 0 denote the undisturbed pressure and density, supposed uniform, and let p, 
p, u denote the pressure, density and velocity at time t for the particles specified by x 0 . 
The equation of continuity is 
and the equation of motion is 
dx 
P te n = P °> 
d 2 x dx _ dp 
dt~ dx n dx 0 
On introducing u, which is dx/dt, these equations become 
du _ p 0 dp du 
dx 0 p 2 dt ’ dt 
1 _ dp 
po 
It is supposed that p is a uniform function of p, and it is convenient to introduce, 
after Riemann, a quantity a- by the defining equation 
d ° = ) V {%) d p 
and the condition that 
0 when 
~+njp- = o, 
ct ex.. 
0 . Then the equations become 
du 
M +II I = 0 ' 
-0 
where II is a function of p defined by the equation 
II 
_ P_ 
Po 
/©• 
The quantity II, which is of the dimensions of a velocity, may be regarded as a known 
function of <x. The value of II when p — p {) is the velocity of sound waves of small 
amplitude in the undisturbed state of the gas. This will be denoted by a. The equations 
are of Lagrangian type, and £ 0 and t are the independent variables, idle quantities 
p and p, like II, can be regarded as known functions of <r. The value of a- when p = p„ 
will be denoted by <r u . 
3. Progressive Waves. — Two quantities r and s may be introduced, after Riemann, 
by the equations 
<t+u = 2 r, (t—u = 2s, 
or 
a — r + s, u — r—s. 
The equations of continuity and motion then give the two equations 
