no THIRD REPORT — 1833. 



an exact differential of a function of three variables, renders the 

 subsequent analytical reasoning much simpler than it would be 

 in the contrary case. This simplification has been proved by 

 Lagrange to obtain in most of the problems of interest that are 

 proposed for our solution *. Euler showed that the differential 

 is inexact when the mass of fluid revolves round an axis so that 

 the velocity is some function of the distance from the axisf. 

 But no general method exists of distinguishing the instances in 

 which the quantity in question is a complete differential, and 

 when it is not. Nor is it known to what physical circumstance 

 this peculiarity of the analysis refers. To clear up this point 

 is a desideratum in the theory of hydrodynamics. M. Poisson 

 has left nothing to be desired in the generality with which he 

 has solved the problem of propagation of motion in elastic fluids ; 

 for in the Memoirs of the Academy of Paris \ he has given 

 a solution of the question, without supposing the initial disturb- 

 ance to be such as to make the above-mentioned quantity an 

 exact differential. His conclusions are, that the velocity of 

 propagation is the same as when this supposition is made ; that 

 the part of the motion which depends on the initial condensa- 

 tions or dilations follows the same laws as in that case, but 

 the part depending on the initial velocity does not return com- 

 pletely to a state of repose after a determinate interval of time ; 

 that at great distances from the place of agitation there is no 

 essential difference between the motion in the two cases. 



III. We turn now to the theory of musical vibrations of the 

 air in cylindrical tubes of finite length. Little has been effected 

 by analysis in regard to this interesting subject. The principal 

 difficulty consists in determining the manner in which the mo- 

 tion is affected by the extremities of the tube, whether open or 

 closed, but particularly the open end. Those who first handled 

 the question reasoned on the hypotheses, that at the open end 

 the air is always of the same density as the external air to 

 maintain an equilibrium with it, and at the closed end always 

 stationary by reason of the stop. The latter supposition will 

 be true only when the stop is perfectly rigid. It does not ma- 

 terially affect the truth of the reasoning ; but if the other sup- 

 position were strictly true, the sound from the vibrating column 

 of air in the tube would not cease so suddenly as experience 

 shows it does, when the disturbing cause is removed ; neither 

 on this hypothesis could the external air be acted on so as to 

 receive alternate condensations and rarefactions, and transmit 



• Mecanique Analyti(]ue, Part II. § xi. art. 16. 



t Memoires de I' Academic de Berlin, 1735, p. 292. 



J torn. X. 1831. 



