of the Sound of Thunder. 39 



article by the assumption of the musical type of the sound-wave. 

 The result shows that all audible musicalsounds are transmitted 

 with the same velocity : and, as the equation of motion in art. 3 

 is linear, it follows that every sound which can be considered as 

 a combination of musical sounds, will be transmitted with the 

 same velocity : and as in general any ordinary function of t can 

 be expanded in the form A' cos (k! t + a!) + A" cos (k' 1 1 + a 11 ) + . . . . , 

 it follows that all ordinary sounds are made up of musical sounds 

 and transmitted with the same velocity. 



10. From this it will be seen that we may consider the equation 



as the definition of the elements of what I have denominated a 

 gentle or ordinary sound. The convenience of analysis requires 

 that, in treating the differential equation of art. 3, we should be 

 able to express Jf t x r in the form of x r multiplied into a constant, 

 or to substitute a constant for the symbol D}; and hence it is 

 obvious there remains yet to be considered the form 



In other words, it remains that we should consider the exponential 

 type x r — A r e kt } which we shall consider to be the type of ele- 

 mentary violent sounds. 



By the substitution of this form in the equation of art. 3 we 

 obtain 



k 2 A r =mf{h) . (A r _!-2A r + A r+1 ) + &c, 



which being a linear equation of finite differences, we find for 

 its solution 



A^Crf'+C'a-*, (l f ) 



the quantity a being such as to satisfy the equation 



k*=mf(h) . (u-d-^ + mffih) . (a 2 -a- 2 ) 2 + ... . (2') 



the form of which indicates that, for every value of k, there is a 

 possible value of a— ex.' 1 , and therefore of a. The case of 

 a— a -1 = is excluded because it makes £=0, and destroys the 

 previous assumption of the exponential type. I assume there- 

 fore the value of a— a -1 to be the measure of the degree of vio- 

 lence of the genesis of any proposed wave. 

 The above leads to the equation 



x = Ce ft * +2rlogea + C'€ w-2rloge<x 



the two terms of which indicate the possible coexistence, at the 

 same point, of two violent waves travelling in opposite directions. 

 We shall therefore only retain the latter term, 



x r =A€ kt - 2rl °s<«, (5') 



as the representative of a progressive wave of the violent type. 



