272 NEWTON S PRINCIPIA. 



and since the pressure varies as the density \ve have 



P = xf, 



and therefore the moving pressure on the element is 



dp, 



K ~ a x. 

 dx 



But the mass moved is D dx ; hence on substitution for p, 

 the accelerating force on the element is 



Thus the force depends on the displacement, and the 

 displacement, in its turn, 011 the force. If we assume a 

 form for the displacement, and then show that this dis 

 placement leads to a force that will produce this exact 

 displacement, we have discovered a possible motion. And 

 if this displacement also agree with all the other conditions 

 of the question, we have discovered the actual motion. 

 For it is clear that from a given disturbance under given 

 circumstances, only one kind of motion can result. Newton 

 assumes accordingly that 



= a. sin (nt mx). 



The accelerating force is then, by two differentiations, 

 K m 2 .a sin (ntmx) 

 = xm 2 . 



that is, the force varies as the distance from a fixed point 

 and urges the particle towards that point. This force is 

 well known to lead to the very form for that we started 

 with, (Prop, xxxviii. of Book I.) provided 

 ?z 2 = x m 2 , 



or the velocity f J with which the wave is propagated is 

 N/x7 This, therefore, is the velocity of sound. 



