359, 360] Propagation of Sound 303 



equations from which the propagation of sound is to be deduced will accord- 

 ingly be the equations of transfer of the four quantities u , v, H and K, 

 corresponding respectively to momentum, mass, and rotational and trans- 

 lational energy. 



From equation (648), the general equation of transfer of any quantity 

 Q, simplified by the suppositions that there are no external forces, that 

 squares of the mass-motion may be neglected, and that the whole motion is 

 parallel to the axis of x, is found to be 



(725). 



To find the transfer of momentum we put Q = u, so that Q = U Q and 

 AQ = 0. We obtain 



^( Wo )=_l(^) = -|-a y c 2 ) = -^-|-(i/K) ......... (726), 



dt 9# 9# 3m ox 



the equation of motion in the simpler theory. 



To find the transfer of mass, we put Q=l, so that Q = l and AQ = 0. 

 We obtain 



the equation of continuity. 



To obtain the equation of transfer of H, we put Q = ^mkW, so that Q= H, 



CH 



uQ =M O H, and AQ is the same as the value of v-j- given by equation (253), 



dt 



namely 



The equation of transfer is accordingly 



iK) ............ (728). 



Similarly the rate of transfer of K is obtained by putting Q = ^mc*, so 

 that Q K, and 



uQ = \tn u (u 2 + v z + w 2 ) 



= \m (M O + u) (w 2 + 2w U + u 2 + V 2 + w 2 ) 

 (3u 2 + v 2 + w 2 ) 



The equation is accordingly 



-|K) ............... (729). 



