30 Mr. E. li. M. Bosancjuet on ilie Theory of Sound. 



C 

 (ii) — —,iimk[vt-\-r), 



("0 - ;:2- 



The expression for the density is (Airv. p. 67) 



Avhich gives for the variable terms deduced from R, 

 (i) __DC^cos%.^-.), 



(ii) -DC-cos/:<r^4-r), 



("0 0. 



The terms of variable density in (i) and (ii) bear the same 



analogy to the first terms in their complete velocities that the 



densities of ordinary sound-waves of transmission do to the 



corresponding velocities ; but if we suppose kr small, these 



changes of density are negligible compared with those that 



would be required to correspond to the surviving terms of the 



C 

 velocities. And ultimately, when the velocity = -^ , there is 



no change of density at all, and w^e may regard the fluid as 

 incompressible under the circumstances. 



This once admitted, in the case where the wave-length is 

 supposed very great in comparison with all the dimensions 

 considered, we can deduce the whole motion very simply from 

 other considerations in the important case where the velocity 



c 



= -^ or there is a uniform floAV — the limiting case when the 



wave-length is infinite. 



Let Sq be the surface of the spherical source, ^^dr the volume 

 of fluid that flows through S^ in the time dt ; then 



dr 



-— = Vn is the rate of flow of fluid through S^, 



dt ^ ta 0? 



^' =V through S. 

 dt ^ 



Since the fluid is continuous and of constant density (accord- 

 ing to the above reasoning), the same quantity of fluid crosses 



