-2 



the Vibrations of an Elastic Fluid, 147 



magnitude of the product 



\ n 1 / \ nj "\ n x ) 



when j9 exceeds n x — \. It has thus been shown that the value 

 x — n x satisfies the equation /=0 approximately, and that the 

 true root differs so much the less from this as x is greater. 

 Ultimately, therefore, i\ \Ze = n v and the condensation is zero in 

 a cylindrical surface of which the radius is r v As n x is any very 

 large integer, we shall equally have r 2 \Ze = n Y + l, r 2 being the 

 radius of the next greater cylindrical surface of no condensation. 

 Hence if D = the common interval between the consecutive sur- 

 faces, D \/e=l. 



Again, the transverse vibrations at great distances from the 

 axis are performed in the same manner, and in the same time, 

 as vibrations along the axis which would result from two equal 

 series of waves propagated simultaneously in opposite directions. 

 In both cases there will be nodes and loops separated by constant 

 intervals. Hence the intervals between consecutive points of no 

 condensation must be the same in both, the times of vibration 

 being the same. As we found that (f> = mcosq(z — a^+c), the 



7T 



interval for the longitudinal vibrations will be, as is known, — ; 



i . $ 



and the interval for the transverse vibrations being — -=, it fol- 



v e 



lows that Ve= — . Or, since V e= s- , we have — = -±, and 



7r Mi 2a 7T 



b 2 4>a 2 

 consequently — z — — g-. The first approximation to the velocity 



of propagation obtained in art. 24 is thus shown to be a\ / \ _j . 



V 7j- 2 ' 



31. By substituting for b 2 and a x from the above results in 



2tt 

 the equations (8) and (9), and putting — - for q, and m! for 



27rm . «• 



r — , it will be found that 



A* 



dd> , . 2-rru! 7r(7r 2 + 4)V 2 4W ^ir 2 (2ir 2 + 7)m! 3 . 6V 



-j- = m sin — — — -TT- 1 cos-—- 10Q a - — sin— 1 ~ 



dz X ba X 128« 2 X 



<=^(l+±) + ^(2^+5). 



Having thus found the forms of the functions (/> and/, and 

 the value of the assumed constant b 2 , the free vibratory motion 

 of an elastic fluid is completely determined, so far as regards 

 vibrations of small magnitude. In another communication I 



