16 PROPAGATION OF LIGHT. 



when it vanishes. In the remaining half of the vibration its 

 values are equal at corresponding times, but with the nega- 

 tive sign, until at length it vanishes again, and the molecule 

 returns to its position of rest, when t - t = T. 



(18) Now, let us suppose that, in virtue of the attraction 

 which subsists among the particles of the ether, the vibratory 

 movement is communicated from the vibrating molecule to 

 the next adjoining, from this latter to the next, and so on. 

 Each molecule will move to and fro, according to the same 

 laws as the first ; but, as the propagation of the vibratory 

 motion requires time, the successive molecules will be in dif- 

 ferent stages of their vibration. The displacements will, 

 accordingly, be expressed by the formula already given ; but 

 the time of the commencement of each will be different, and 

 proportionate to the distance traversed. Let x be the dis- 

 tance of any molecule from that corresponding to the origin 



tJC 



of the time, and v the velocity of- propagation ; then t = - . 

 Also, if A be the length of an undulation, or the space tra- 

 versed in the time 1 

 the formula becomes 



versed in the time T, T = -. Making these substitutions, 



. ; , . ^ 

 y = asm -r- (vt-x). 

 A 



When x is constant, this equation gives the relation be- 

 tween the displacement and the time, or the law of the 

 vibratory motion, for any one molecule whose distance from 

 the origin is given. On the other hand, if t be constant, 

 the equation gives the relation between the ordinates and 

 abscissae of the consecutive molecules, for a given time, or the 

 equation of the wave.* 



* If we take the partial differentials of this equation with respect to t and x 

 we see that 



ftL-fclgift 



d? dz* ' 



