280 Mr. Challis on the Theory of the 



whatever be t, and a certain value I' for vi'hich 



F(l' - at) =- F{1' + at), 

 whatever be t. The motion would be equally determined by the 



equations, 



V = F{x- at) + F{x -\- at), 



as = F{x — at) — F{x + at), 

 the origin of x being- changed; but the former are the more 

 convenient equations, because s does not change sign with the 

 change of the direction of propagation, whereas v does. At the 

 point, the abscissa of which is I, v = o, and as = 2F{1 — at). This 

 point is called a node. The other point, whose abscissa is /', 

 is called a loop, and at it as = o, v = 2F(l' - at). It is plain 

 that the loops as well as the nodes recur as often as the cui"ve 

 cuts the axis, and that each loop is separated from the adjoining 



node by the constant interval - . 



4. According to a remark before made, the preceding rea- 

 soning, being conducted without reference to the manner in 

 which the fluid was put in motion, must be considered in a 

 general point of view. The inference to be drawn from it is, 

 that the motions of the particles in general, result from two 

 motions of propagation obtaining simultaneously in opposite di- 

 rections, that the velocity of propagation is always equal to the 

 constant a, and that, considering the propagation in one direction 

 only, the motions of the particles are in all cases primarily, (not 



necessarily,) such as are indicated by the curve y = m\ sin — ; 



in other words, their motions are always resolvable into a motion 

 of this kind. The integral of a partial differential equation 

 .should always be subjected to a discussion like the foregoing, 

 before any application is made of it, and for the purpose of 

 directing us to the mode of making the application. Another 

 remark, which it is important to make, is, that although the 



