the Vibrations of an Elastic Fluid. 145 



possible position of the above-mentioned plane. By performing 

 the summation, substituting r~ for x~ -f y 2 , and determining a so 

 as to satisfy the condition that/=l where r = 0, the result is 



eV e s r 6 



/=! er + l2 o2 t 2 92 02 "•" ^C' ■ • C-^J 



. <v 



2 2 1 2 .2 2 .3 2 



This value of/, containing no arbitrary quantity whatever, ex- 

 presses the general law of the arrangement of the condensation, 

 in the case of free motion, about the axis of rectilinear motion. 

 It is evident that the same result might be obtained from (10) by 

 supposing, as might be presumed to be the case in free and un- 

 disturbed motion, that the condensation a, and therefore/, is a 

 function of r. I have pointed out other methods of obtaining 

 the series (12) in the communication of December 1852 before 

 referred to. 



29. To ascertain the precise character of the free vibrations of 

 an elastic fluid, it will now be necessary to discuss the properties 

 of the function /. First, it is to be observed that, since the con- 

 densation in any plane transverse to the axis of z is proportional 

 to/, those values of r which make /vanish are radii of cylin- 

 drical surfaces in which the condensation is zero throughout the 



motion. Similarly, those values of r which make ~ vanish are 



J dr 



radii of cylindrical surfaces in which the transverse velocity is 

 always zero. In order to calculate the magnitudes of these dif- 

 ferent radii, it is necessary first to determine the value of the 

 unknown constant e. I have given a process for effecting this 

 determination in a communication to the Philosophical Magazine 

 for February 1853, under Prop. XL; and no other method of 

 obtaining the same result has since occurred to me, although I 

 can scarcely doubt that the problem admits of being solved in a 

 different manner. At the same time I have seen no reason to 

 call in question the exactness of the solution there given, which, 

 as it is important, and admits of being exhibited in a somewhat 

 simpler form, I shall now repeat. 



30. Assuming/to be a function of?-, the equation (10) becomes 



and the equation (12) is the integral of this equation in a series. 

 The above-mentioned process for finding the value of e requires 

 the determination of values of r which make / vanish at very 

 great distances from the axis ; and at first sight it might be sup- 

 posed that for very large values of r the middle term of the above 

 equation might be omitted, and the function that/ is of r be ob- 

 tained with sufficient approximation by integrating the equation 



