268 The Rev. J. Challis on the analytical Determination 



2 /, be deranged from its quiescent state and put into any ar- 

 bitrary shape, subject, however, to the condition that each 

 point is very little removed from the place it would have had 

 at rest, and no two consecutive elements of the chord make a 

 large angle with each other. The deranged portion being held 

 a while in its new position, it is required to find what will 

 ensue when it is left to itself. Let y be the distance of any 

 point of it from the horizontal line in which the chord rests, 

 and X the distance of the same point, measured horizontally, 

 from the middle of the derangement, t the time measured 

 from a given instant. The known partial differential equation 

 applicable to this case is 



where a^ = */ "-Zgc, g being the measure of the force of 

 gravity, and c the length of the chord whose weight measures 

 the tension. The integral of this equation is 



y = F(a;-al) -\-f(a:-\-at), (1.) 



from which is derived i 



-^ = — aY' {x—at)+af' (-^x + at), (2.) 



a V 



an expression for the velocity in a direction perpendicular to 

 the horizontal line of abscissae. We may consider it as proved 

 by Lagrange, that in assigning any particular forms to the 



d y 

 functions which express the initial magnitudes of 3/ and — — , 



we are at liberty to take only such values of them as corre- 

 spond to values of x lying between arbitrary limits, and may 

 suppose the forms of the functions to be quite different for 

 different intervals along the line of abscissae. Hence the 

 equations (1.) and (2.) will apply to the case of derangement 

 supposed above, and we may say, for instance, that the de- 

 ranged portion of the chord shall take the form given by the 



equation y = h l\ ^j from x=— ltox~-\-h with- 

 out considering any of the values of ^ that result from other 

 values of x. The kind of motion that takes place when the 

 chord is abandoned to itself would be inferred as follows, ac- 

 cording to the received method, an instance of which may be 

 seen in the Treatise on Sound in the Encyclopcodia Metropo- 

 litaiia, Arts. 57 — 65. 



Let t be dated from the instant the motion commences, and 



