ON LAGRANGE’S BALLISTIC PROBLEM. 
173 
The two equations containing Ap and A u give 
A p = p 0 w A u 
and 
p A u + u A p — p 0 wu A u + p 0 w (p/p 2 ) Ap. 
The terms uAp and ppvuAa in the second of these equations cancel, and then, by 
eliminating A u between the two equations, we find 
9 p 2 Ap 
w = 2 pr~ 
Po &P 
Since there is no actual discontinuity and p is a uniform function of p, we may replace 
Ap/Ap by dp/dp, and thus obtain the equation 
iv u = 
p 2 dp 
p 2 dp 
which shows that the velocity of the junction relative to the medium is that which 
was previously denoted by 11 . 
If motion is set up in one part of the gas, and advances into previously undisturbed 
gas, the value of p at the junction is po , and therefore the velocity of the front of the 
wave, relative to the medium or to the tube, is that which has been denoted by a. 
5. Nature of the Motion in a Compound Wave .—Important results can be obtained by 
regarding x 0 and t as functions of r and s. On interchanging the dependent and inde¬ 
pendent variables in the equations 
— + JT— = 0 — -II — =0 
dt 0X O ’ dt dx 0 
we obtain the equations 
dX 0 _ TT ft _ Q 
05 05 
0X° + n^ = 0 . 
dr 
dr 
Now the differentials of x 0 and t are always connected with those of r and s by the 
formulae 
dx 0 = 
dx 0 
dr 
dr+ d -^ds, 
05 
dt = ? ^-dr+^ds. 
dr 05 
Hence the places in the medium, and the times, at which any particular value of r is 
found, vary according to the formulae 
dx 0 = ^ ds = II ^ ds, 
05 05 
dt — ds — 
05 
and thus it appears that any value of r is transmitted through the medium, in the 
direction of increase of x, with the velocity II. In like manner it can be shown that 
any value of s is transmitted in the opposite direction with the same local velocity. 
