Small Vibratory Motions of Elastic Fluids. 319 



and for simplicity put = tan §. It will be found then that, 



—- = — Ul — e - ) cos 6 sin ( \- h) - 2e ■ sin d sin 



at p ^ V X -' 



vaty 



The first term in the brackets, commencing from o will quickly 

 increase, and the other, always small by reason of the factor 

 sin" I, will rapidly decrease, and the vibrations will soon become 

 isochronous. Also they may be of considerable magnitude, be- 



m 



cause the ratio — is not in general small. After a short time, 

 V 



ds m ^ . iTzat 5,\ 



-77 = — cos d sin ( — he). 



at p \ X / 



V 



The quantity n depends on the length of the resounding chord, 

 or a submultiple of its length. Tlie cause producing the waves 

 may be a vibrating chord, the length of which is equal to that 

 of the other, or a submvdtiple of its length, or lastly a multiple 

 of it, and in this last case the resonances are caused by the 

 harmonic waves of the vibrating chord. 



On similar principles would have to be calculated the eftect 

 of a series of waves, propagated along the interior of a musical 

 pipe, in producing those vibrations of it, which determine the 

 timbre or quality of the note, and which I suspect are prin- 

 cipally concerned in fixing the value of X. (See Art. 13.) 



21. In conclusion I will state the inference to be drawn 

 from all that has gone before respecting the application of the 

 integrals of partial differential equations to physics. 



As the integral of a partial differential equation contains 

 arbitrary functions, if we consider it in a purely analytical sense, 

 it may be satisfied by any one of the infinite number of functions 

 we can form at pleasure, or by any number of them combined, 

 or lastly by the combination of portions of any number of them, 



