ACTED UPON BY THE VIBRATIONS OF AN ELASTIC MEDIUM. 347 



e ' fe r <p'(t)dt, 



and therefore the same as if the velocity were simply m<j>(t). Hence 

 a small sphere moving with a uniform velocity suffers no resistance; 

 and if its velocity be partly uniform and partly variable, the resistance 

 depends only on the variable part. 



9. I come now to the consideration of the motion of a small sphere 

 supposing it acted upon by the pressure resulting from a series of 

 vibrations of the fluid, no other force acting. I suppose the vibrations 

 to be propagated with the uniform velocity a in the positive direction, 

 and the velocity of the vibrating fluid to be m<p(t) at the origin of 

 x -at any time t. At the same time t at any distance x from the 



origin the velocity is <p [t ■) , being that which was at the origin 



at the time it ) . Suppose the centre of the sphere to be at the 



origin of x ' when t = 0, and to be at the distance x at the time t. 

 For the sake of simplicity I shall first assume the vibrations of the 

 fluid to be unaccompanied by change of density, which is a supppsable 

 case if we conceive all the parts of the fluid to move in the direction 

 of x at the same time with the same velocity. Now it is clear that 

 the action of the fluid on the sphere depends only on the difference 

 of their velocities. And the mathematical conditions of the question 

 will remain the same if we suppose the fluid to be at rest and the 



sphere to have the velocity — \mfpit ) — -tt } • Calling this velocity 



— mx(t), and f the resulting accelerative force of the sphere, we shall 

 have by what was proved in Art. 5, 



- ami \ -2* r ii -al\ 



J= ~-\e T Je'^{t)dt-ke '). 



Let us now suppose the fluid vibrations to be accompanied by change 



of density. If pi be the density where the velocity is m<p It J , it 



-<t>(t- x \ The 

 is well known that we have the exact relation pi = e a N «/' 



