232 Mb HOPKINS ON AERIAL VIBRATIONS 



different solutions which mathematicians have given of the problem in 

 question. The principle on which we ought to proceed in making such 

 assumptions is obvious ; they should be subjected to no restrictions, 

 (not imposed on them by our theory), which are not necessary to draw 

 those deductions and inferences from our mathematical results^ which 

 admit of verification by experiment, to the test of which an assumption, 

 in any degree arbitrary, must necessarily be subjected before it can claim 

 our confidence. The physical conditions however on which the solutions 

 of this problem depend, (as far as it is distinct from that of the motion 

 of a wave along a uniform tube of indefinite length), have neither 

 been assumed on this principle, nor subjected, as far as I am aware, 

 to this experimental test. It has been principally with the view of 

 remedying these defects that I have prosecuted the researches, an account 

 of which I have now the honour of laying before the Society. 



1. The physical conditions assumed by Euler, and by most of those 

 who have since written on the subject, are, that the particles of air at 

 the extremity of a closed tube are always at rest; and that no con- 

 densation of the air takes place at the extremity of an open one. The 

 first condition involves the supposition of the perfect rigidity of the 

 material with which the tube is stopped. This cannot be accurately 

 true, but probably leads to no error very appreciable to observation. 

 The second condition assumes an eqviality in the densities of the external 

 air, and of that within the tube immediately at its open extremity, 

 during the whole time of the vibrating motion, in the same manner as 

 if the air were at rest. This supposition carries with it but little 

 appearance of being even very approximately true; for it is difficult 

 to conceive how a sonorous wave could thus be produced and maintained 

 in the surrounding air from the open extremity of the tube, and it 

 appears perfectly irreconcileable with the fact of the sudden cessation 

 of sound after the cause producing it has ceased, M. Poisson, struck 

 with these objections, has assumed another physical condition as appli- 

 cable to any tube, whether open or stopped, viz. that there exists at the 

 extremity of the tube, during the whole motion, q constant relation 

 between the velocity of the particles of the fluid at any instant, and 

 its condensation, this relation depending on the nature of the substance 



