ON LAGRANGE’S BALLISTIC PROBLEM. 
175 
These are the relations of duality familiar in discussions of partial differential equations,* 
and we may put 
dx 
z 
x 0 
, dx 
dx 0 +t dt ' 
Actually Z could differ from the right-hand member of this equation by a constant, 
but as such a constant woidd be irrelevant, the above will be taken as the relation 
between Z and x. 
The equation satisfied by Z can be written in either of the forms 
or 
0^z (~Ldii)dz_vz 
da- 2 \II Cla) 3<x 3 U 2 
d 2 Z fJ_dll)fcZ 3Z' 
dr 05 \2II d<rj\dr ds / 
- 0 . 
7. Relation between Pressure and Density .—The analysis of the problem is not rendered 
more difficult if the adiabatic relation between pressure p and volume v is taken in 
the form p (v — b) y = const, instead of the more ordinary form pv y = const., and 
the former is more suitable for the applications which we have in view. We shall 
accordingly take the relation between pressure and density to be 
IV 
d> 
p[ -~ } = p % r ~ d ) 
i iV 
d —pi I_ 
Po fi-p 
\(y— 1)/2 
where (3 and y are constants. Then the following results can be obtained without 
difficulty :— 
_ * 2 J Poy(fi-po) ' 
y— 1 I. fipo 
9 ’ i p„y (ft—p<>) 
dpo 
Tn 
— 1 
a 
_ f po yd 
Pn{d — Po)j 
P = P» (o-/o- 0 ) 2n+1 , 
II = a(o-/o- 0 ) 2n , 
where 2 n has been written for (y + l)/(y — 1). 
The equation for Z can now be written 
S 2 Z . 2 n cZ b 2 Z 
\ ^2 
)(T OU 
+ 
or 
9 2 Z 
dr ds 
+ _»_ (P + 3Z'| = 0. 
r + s\dr os. 
* The reduction of the equations governing the propagation of plane waves of finite amplitude to a 
single partial differential equation of the second order was effected byRiEMANN, who worked with “ Eulerian 
equations. The use of the principle of duality to connect Z and x was noted by Hadamard. 
C 
VOL. CCXXII.—A. 
