Small Vibratory Motions of Elastic Fluids. 293 



I' 

 fluid ; then the mean velocity it receives is — , and the mo- 

 mentum communicated to it 



2 

 Hence, Mw = {,x + M) F+ m"ln V", 



Mw- {M + m) f^ 



n = 



2 /,» W 



and^ = ^(M^-i»/ + .) 



There woidd probably be great practical difficulty in determin- 

 ing f^, but the difficulty might be diminished by choosing m" 

 very large, which may be done without affecting the accuracy 

 of the determination. 



10. Because w is a function of x and t, 



(dv\ dv dv dx 

 \dt) "^ ~dt "^ dx ' dl' 



Let the propagation be in a single direction : then 



r/ A\ dv „, . . dv 



v = F{x — at); -j- =^ — a .F' {x ~ at) = - a-r-- 



XT /dv\ dv / v\ „ dx 



"^"^^' {Tt)^dt{'-a)' ^''di = ^- 



V 



But - may be neglected in comparison of i, as has been done 



before. Therefore the accelerative force acting on a particle at 

 the distance x from the origin, very nearly 



_dv _ d^(f> 

 dt dtdx' 



But is by hypothesis very small, and the reasoning is not 

 accurately applicable to cases in which it exceeds a certain 



