NEWTON S PRINCIPIA. 267 



moving across the field, because everything is fastened to 

 the ground ; but there is the appearance as if something 

 were moving along. Each particle of corn in turn descends 

 and rises again, and, as each ear is a little later in its mo 

 tion than the one in rear, the state or position in which 

 the first was, has travelled to the second. Our first idea 

 of a wave is, that it is a state of motion which travels 

 along, the particles themselves only oscillating. Our 

 second, that this state of motion may gradually change, 

 and either increase or die away. 



Suppose we have a series of particles in a straight line, 

 let them begin to move up and down, either in this straight 

 line, or in any parallel straight lines, so that if one be taken 

 whose mean position, measured along the straight line is 

 distant x from the origin of measurement, its distance at 

 the time t from its mean position is represented by the 

 formula 



y = af(n t m x), 

 where /is any functional symbol. 

 Then here, so long as 



n t m x 



is constant, y remains the same ; that is, a &quot; state &quot; travels 

 along in such a manner that its distance x at any time t 

 is given by this formula. 

 Differentiating we have 



dx 



but - is the velocity ; hence the &quot; state &quot; travels with 



the velocity - ; this, therefore, is the velocity of the wave. 



Suppose we call this v, then we have 



?/ = af m (y t x}. 



