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X. On Aerial Vibrations in Cylindrical Tubes. By William 

 Hopkins, M.A. Mathematical Lecturer of St Peter's CoUege, 

 and FeUow of the Cambridge Philosophical Society. 



[Read May 20, 1833.] 



The problem which has for its object the determination of the 

 motion of a small vibration propagated in an elastic medium along a 

 prismatic tube of indefinite length (the motion of every particle in 

 each section of the tube perpendicular to its axis being the same) was 

 long since solved by Euler and Lagrange. The problem, so nearly 

 allied to this — to determine the motion of an aerial pulsation in a tube 

 of definite length — has not been so satisfactorily solved, the tube being 

 either open at the extremity or stopped with a substance possessing 

 some degree of elasticity. In addition to the difficulties of the former 

 problem, we have in this latter one those still more formidable difficulties 

 which exist in the determination of the circumstances of the motion 

 at the confines of two elastic media in the closed tube, or at the 

 extremity of the open one, where the air in the tube communicates 

 with the circumambient air. These motions must no doubt be deter- 

 minable from the nature of the media, and the causes producing and 

 maintaining the vibrations, having nothing arbitrary in them, except 

 what may be so in the original disturbance ; but I am not aware 

 of any progress having been made in the direct solution of these 

 questions, which now forms one of the greatest desiderata in the appli- 

 cation of mathematics to physical science; and in our inability to 

 determine these motions at the extremity of the tube, either by theory 

 or direct observation, we are driven to the necessity of assumptions. 

 It is from a difference in these assumed conditions that we have the 



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