On the Theoretical Value of the Velocity of Sound. 285 



Therefore the waves cannot be supposed to be spherical. 



That there may be no excuse for misapprehension as to the 

 result attributed to the analysis in the second member of the 

 syllogism, I proceed to exemplify that result by a numerical 

 instance. The pressure being a^(l +5), the value of the con- 

 densation s at any distance r from the centre, and at any time 

 ti is admitted to be given by the equation 



Yir-at) 



s— — -. 



r 



Since the function F is arbitrary, it may be supposed that 



a . 27r , ^ , 



5=-sm — {r—at-\-c), 



[Xf X, and c being certain constants. It is also admitted that 

 the function F may be taken discontinnously, that is, from one 

 zero value to another zero value; and that all other values of 

 s not included between those limiting values may be zero. 

 Let therefore the values of the circular function be taken from 



r—ati + c=0 to r—at^-\-c= -. Then, the mean density of 



the medium being unity, the quantity of condensed matter in 

 the space occupied by the wave above matter of mean density 

 occupying the same space, is the integral o^ ^irr^sdr taken be- 

 tween the limits just mentioned. Call this quantity «, and 

 for the sake of definiteness of conception, let the fluid under 

 consideration be contained at the time /j between two rigid 

 spherical surfaces, the radius of one of which is 1000 feet, and 

 that of the other 1,000,000 feet. There is nothing in the an- 

 tecedent investigation to exclude such a supposition, and for 

 the purpose of the argument these numbers will serve as well 

 as any others. Let the fluid of mean density which would 

 fill the space between these surfaces be A in cubic feet, which 

 of course is a constant quantity. Then a. being expressed in 

 cubic feet, the whole quantity of matter at the time ^1 is A + a. 

 To express a numerically let X=\ foot, and let the constant 

 /u.= 1, which amounts to supposing that the maximum conden- 

 sation at a distance of 10,000 feet is 0,0001. Consequently 



« = 47r / ( 



Ir sin — {r—ati + c)dr 



the exact value of which integral between the limits r—at^ 



+ c=0 and r—at^-\-c=- is 4Arj, 7\ being the distance of the 



maximum condensation from the centre at the time/j. Hence 

 whenr, = 10,000feet, thewhole quantity of matter is A +40,000 



