290 Mr. Challis on the Theory of the 



But as the order of condensations at a given point is reversed 

 when the direction in which the curve travels is changed, the 

 succession of the velocities which a given particle undergoes, 

 must also be reversed. At the .same time the direction of its 

 motion is changed. Now that this may take place, will appear 

 by considering that iifp = x, pq=y, the accelerative force acting 



on a particle situated where the condensation is y, varies as ^, 



and is therefore the same, and directed in the same way in the 

 two propagations. Hence, / being the same function of s in the 

 two motions, we shall have 



an equation embracing both motions, and shewing that for the 

 same value of s, v has two equal values, one positive, the other 

 negative, one belonging to the motion forwards, the other to 

 the motion backwards. This being premised, let a diaphragm 

 be made to move so as to generate the wave fend, and when 

 the portion fenc is generated, let it be suddenly stopped. The 

 wave will in consequence assume the type fe'n'nef. And it may 

 be shewn precisely as before, that a particle may perform the 

 motion indicated by the portion n'n of the primary wave, what- 

 ever be its prior and subsequent motions. 



Suppose now any number of curves to be described, the 

 general equation of which is, 



y = m\ sin — ^- — - + m\ sin — ^-; — - +m \ sm ^ , — - + &c. 



and which differ according to different hypotheses made on 

 VI, \, c, &c. Then it follows from what precedes, and from the 

 principle of the co-existence of small vibrations, that if any 



