304 Mr. Challis on the Theory of the 



side holes just as they do at the extremity, and thus it is that 

 the waves which are separately generated in the external air 

 at the side holes and at the extremity, produce sounds of the 

 same pitch and quality. It would be difficult to explain this fact 

 according to the received theory. 



Upon the whole, our theory seems to prove that the scries 

 of notes is attributable to the nature of the fluid, but that the 

 particular value of \ is due to causes, independent of the fluid, 

 which are not yet satisfactorily understood. 



III. .Motion in Space of two and of three Dimensions. 



14. Next let the motion take place in space of two dimen- 

 sions. The equation for this case is, 



dt- \dx- dy- ) 



dy- 



. , o d(t> d<b d<t> 



And a-s+-l = o, p = -^, q=^, 



p and q being the velocities in x and y respectively. To inte- 

 g;rate the above equation, suppose x-+3^' = r^ Hence 



d<f> d(p X 

 dx dr ' r' 



dr<p _ <f (/) x° d(j) /I ar'\ 

 dx" ~ dr'' ■ r- dr \r r) ' 



^ = ^ -E + ^ /i _ ^\ 

 dy^ dr ' r dr \r r'/ ' 



dr(p d'<p d^<p d<p 

 dx- dy^ ~ dr° rdr ' 



