40 On ike I 'd»iity of the Sound of Thunder. 



From this equation wc determine, as before, the velocity of a 

 violent sound to be • 



•-nr^vr^-' {/'(A).(«-»-') 9 +/'(2A)-(« 9 -«- a ) 2 +-}i, (7') 



Z log e a 2 log e a 

 ■ result which may be written in the form 



i i 2 / « + «-' V L 1 Z^+l + a-V, 

 1 + 2n*~2~~V + 3 2 \ 3 / +••• 



V~2loi^ 



l+ 29 + g 2 



V being the velocity, before determined in art. 8, for all gentle 

 sounds. That the full meaning of this equation may be obvious, 

 it is necessary to remember that 



21og e « "" 1+ TT273' + 1.2.3.4.5 +,, * * {) 



11. It is manifest, from inspection of the last two equations, 

 that when « = 1 or a— a -1 =0, thenr = V. This case therefore 

 connects (or, rather, stands between as the limit of both) the 

 gentle and the violent sounds. But for every value of a greater 

 than unity, we perceive that v must necessarily be greater than 

 V, which is the limiting velocity of gentle sounds, for both the 

 factors of the right-hand term of equation (8) are greater than 

 unity. 



12. In art. 8 we found an expression for the velocity of 

 transmission of ordinary sounds which was independent of the 

 pitch. But equation (8) we perceive involves a— a -1 , the degree 

 of violence belonging to each wave : and hence it is not true of 

 violent sounds that they all travel with the same velocity. The 

 elements of a compound violent sound will also not keep together 

 unless the value of a be the same in all, or vary continuously 

 from one element to another. In the latter case the sound will, 

 by the unequal motion of its elements, have a tendency to 

 lengthen ; and if a do not vary continuously, the sound will soon 

 be heard, not as one clap, but as a series of minor claps, or as a 

 rattle. 



13. It is seen from observing the forms of equations (8) and (9), 



that such a value can be found for a as shall make the ratio ^ 



as large as we please. It would appear therefore that there is no 

 other limit to the velocity with which a violent sound is trans- 

 missible through the atmosphere, than that which the possibility 

 of supplying a sufficient degree of force in its genesis may impose. 



(8) 



