Mathematical llieory of A'&tnal Vibrations. 89 



step of the reasoning by which the conclusion was arrived at, 

 the proof being a reductio ad absurdum. I shall therefore 

 confine myself to the single objection of this nature brought 

 forward by Mr. Stokes in his communication to the Philoso- 

 phical Magazine for January, viz. that which he expresses by 

 saying that I "integrate through an infinite ordinate." Now 

 the very terms of this objection preclude the necessity of 

 answering it. For it admits the existence of an infinite ordi- 

 nate, and consequently of some valid reasoning by which it is 

 shown to exist. I contend for nothing more. The expres- 

 sion "infinite ordinate" is a symbolical term to indicate that 

 a finite change of velocity or density occurs in an infinitely 

 small space. I have expressed the same thing in plainer 

 terms by saying, that the same point of the same wave is at 

 the same time a position of maximum velocity and of no velo- 

 city. It is clear, therefore, that there is no question as to the 

 mathematical reasoning, but as to the interpretation of the 

 result. I consider that the infinite ordinate is condemnatory 

 of the hypothesis of plane-waves: Mr. Stokes says that it is 

 indicative of certain physical circumstances analogous to a 

 bore. As this difference is not likely to be removed by fur- 

 ther mathematical discussion, I pass on to the case of spherical 

 waves. 



First, I propose to exhibit as distinctly as possible the rea- 

 soning which Mr. Stokes has undertaken to call in question. 

 For this purpose I shall commence with the usual hydrody- 

 namical equations applicable to small disturbances, and pro- 

 ceed through the different steps in order, numbering the pa- 

 ragraphs for the sake of reference. 



(1.) Formation of a general differential equation applicable 

 to small condensations. 



Let/> = a^(l +5), x> being the pressure and s the small con- 

 densation at the time / at a point whose co-ordinates are a;, y^ z. 

 Then if 11, v, w be the resolved parts of the velocity at the same 

 point and at the same time in the directions of the axes of co- 

 ordinates, we have the known approximate equations, 



c,ds_ djt, „^di dv_ ^^ds_ dw_ 



""dx^dt^^' ""'dy^dt-^' ""'dz^dt-^' 



ds du dv dw _ 

 dt dx dy dz~ ' 



By differentiating the last equation with respect to /, we 

 obtain 



dt"^ "^ dtdx ■*■ dtdy "^ dtdz ~ ' 



