Mathematical Theory of A'Mal Vibrations. 91 



represent respectively the condensations 5j, 53, and OM = r,, 

 OM'=r2, wehave ^ = ^. 



(5.) The contradiction. 



Draw an ordinate })m indefinitely near to PM, and />W 

 indefinitely near to P'M', and let the interval Mm=M'm' = a. 

 Then the quantity of matter contained between the spherical 

 surfaces of which the radii are /-j and r^ + u, beyond what would 

 exist in the same space in the quiescent state of the fluid, is 

 4Tr,^5,a; and similarly the quantity of condensed matter be- 

 tween the surfaces of which the radii are r^ and r^ + a is 

 4>Trr^\ci. The same reasoning applies to every set of corre- 

 sponding ordinates of the two curves. Hence by the prin- 

 ciple of constancy of mass employed in investigating one of 

 the general hydrodynamical equations, those two quantities 

 must be equal to each other; that is, A!yrr^s^ot. = ^%r^^s,^. 



ST* 



Consequently — = -\. This result, which is incontrovertible, 



is at variance with the conclusion in (4?.). I infer, therefore, 

 that the hypothesis of (2.) is inadmissible. 



I have been thus explicit for the purpose of stating distinctly 

 the course which this discussion must take. It will be observed 

 that subsequently to making the hypothesis of (2.), the velocity 

 of the fluid is nowhere introduced. The rules of right rea- 

 soning absolutely forbid the introduction of any expression 

 for the velocity by an opponent, simply because such expres- 

 sion cannot be obtained without adopting the very hypothesis 

 the legitimacy of which is the point in dispute. Mr. Stokes 

 from beginning to end has argued from a value of the velocity 

 derived from the disputed hypothesis. To all his argument 1 

 have therefore this one answer : the truth of the expression 

 for the velocity is not proved. It is quite necessary that the 

 discussion should turn on the reasons which I have alleged 

 for the different steps of my argument, without taking for 

 granted the legitimacy of the hypothesis of spherical waves. I 

 am discharged from the necessity of dwelling on the details 

 of Mr. Stokes's reasoning, because the whole of it contains a 

 petitio 2>rincipii. Indeed I should desert the position which 

 the acknowledged rules of right reasoning compel me to take, 

 if 1 made a single remark which implied an admission that 

 that reasoning required from me any answer. 1 proceed, 

 therefore, with the subject which more strictly accords with 

 the title of this communication. 



I shall commence by saying that I regret Mr. Stokes does 

 not feel himself at liberty to give me the benefit of his stric- 

 tures on my mathematical theory of ray-vibrations, and to 



