Small Vibratory Motions of Elastic Fluids. 315 



m undergoes just as the wave leaves the mouth, but considera- 

 tions like those of Art. 11. might enable us to do it. The 

 constancy of vis viva must at least approximately obtain, and 

 this alone is sufficient to shew that a series of waves equal to 

 the incident cannot be reflected at an open mouth in the way 

 D. Bernoulli supposed. 



If a trumpet-shaped tube, the form of which is a surface, 

 of revolution be fitted to the cylindrical tube, the boundary of 

 that portion of the wave which emanates from its surface, will 

 be a surface generated by the revolution round the axis of the 

 tube, of an involute of the curve which generates the surface 

 of the trumpet; for the surface of the wave, where it meets the 

 surface of the trumpet, must be perpendicular to it, since the 

 direction of the motion of the particles, which is always at right 

 angles to the former, must be along the latter. The condensa- 

 tions will be greatest in the direction of the axis, as well from 

 the manner in which the wave spreads laterally, as because in this 

 direction the curvature of the wave is least. Hence a trumpet 

 sounds loudest in the direction of its axis. 



It has been remarked by Euler, that no stop-pipe is musical 

 unless it be cylindrical ; and the fact may be explained by what 

 was said in Art. 13, where it appeared that the density of the 

 air at the open extremity must always be the mean. This 

 cannot be if the tube be of any other shape than that of a 

 cylinder, because the line of propagation from the mouth to the 

 mouth again, will be less along the axis than in any other di- 

 rection, if the closing end be a plane perpendicular to the axis. 

 But if it be made to fit the boundary of the wave, theory shews 

 that the pipe may be musical. 



20. Because v = as for motion in space of three dimensions, 

 the resistance which a projectile whose velocity is small in com- 

 parison of a, encounters from the pressure of the air varies as 



RK2 



