ON LAGRANGE’S BALLISTIC PROBLEM. 
169 
In tlie problem* * * § it is supposed that a given mass of gas, which is initially in a uniform 
state, is contained in a segment of a tube of uniform section. At one end the segment 
of the tube is bounded by a fixed transverse section, and at the other end the tube is 
closed by a piston of given mass, which is initially at rest and is free to move along the 
tube without resistance. It is required to find the subsequent states of the gas and 
the motion of the piston. 
Under the pressure exerted by the gas the piston begins to move, and wave-motion 
of finite amplitude is set up in the gas. The waves are plane. The theory of plane 
waves of expansion of finite amplitude has been the subject of much study,f chiefly 
in connection with the question of the initiation and maintenance of surfaces of 
discontinuity. The difficulties associated with this question do not arise in Lagrange’s 
problem, because the waves that are generated are always waves of rarefaction, and 
there is no tendency to discontinuity in waves of this type. Among the results that 
have been obtained in the theory of plane waves of finite amplitude, two are specially 
important for our present purpose. The first of these is that there exist waves of the 
type known as “ progressive waves,” and that they are the only ones that can advance 
without discontinuity into gas at rest. They are sometimes described as “ motions 
compatible with rest.”J The second important result is that the equations governing 
the propagation of waves which are not compatible with rest can be integrated.§ Such 
waves will be described in the sequel as “ compound waves.” 
The most important writings in which Lagrange’s problem is dealt with are the 
memoir of Hugoniot cited above, H. Hadamard’s ‘ Legons sur la Propagation des 
Ondes,’ Paris, 1903, and a memoir by F. Gossot and II. Liouville in ‘ Memorial des 
Poudres et Salpetres,’ vol. 17, 1914, p. 1. 
The problem is not rendered essentially more difficult if it is supposed that the 
segment of the tube occupied by gas is bounded by two movable pistons of given 
masses. Provision can be made for the case of a fixed end by taking the masses of 
the two pistons to be equal, for then there is never any velocity at the section midway 
between them. 
The tube will be thought of as running from left to right. When the pistons begin 
to move progressive waves set out, one from the left-hand piston with a front proceeding 
towards the right, the other from the right-hand piston with a front proceeding towards 
the left. These waves meet at the middle section, and from that section there then 
sets out a compound wave, which has an advancing front, proceeding towards the right, 
and a receding front proceeding towards the left. This wave will be described as the 
* S. D. Poisson, “ Formules relatives au Mouvement du Boulet . • • extraites des Manuscrits de 
Lagrange,” Paris, ‘ J. Ec. Pol.,’ call. 21 (1832). 
f Reference may be made to Lamb’s ‘ Hydrodynamics,’ cli. 10. 
f H. Hugoniot, Paris, ‘ J. Ec. Pol.,’ call. 57 (1887) and call. 58 (1889). 
§ B. Riemann, 1 Gottingen Abli.,’ vol. 8 (1859-60) ; also £ Ges. math. Werke,’ Leipzig, 1876, 
p. 145. 
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