Mr Sang's Analysis of the Vibration of Wires. 313 



which epoch, since x and y are equal respectively to a and b, 



the velocity of the trajectile is zero. Thus the equations 



Iff t y w r 



^ = a cos y^, y = bcos ^, 



represent a motion caused by drawing the wire aside, and then 

 abandoning it to the effects of its own elasticity, in which case 

 comparative and absolute magnitudes of a and b are determined 

 by the direction and extent of the primitive deflection ; of these 

 the essential characters of the trajectory are entirely indepen- 

 dent. But if an impulse be given to the wire at the instant 

 of its discharge, the time v has some finite magnitude upon 

 which the shape of the curve essentially depends. On account, 

 however, of the imperfections to which all adjustments are liable, 

 it will be seen that v passes gradually through every assignable 

 value ; and thus all the varieties of the trajectory described by 

 a given wire can be exhibited by drawing it aside and abandon- 

 ing it. 



For the delineation of the curve we have the following simple 

 geometric construction. 



Round the point of rest let two straight lines, whose lengths 

 are a and b, turn uniformly in the times 2 T and 2 U : let one 

 line constantly parallel to Y pass through the extremity of a, 

 and another parallel to X through the extremity of 6; the in- 

 tersection of these two lines will trace out the curve. 



Perhaps a more convenient construction may be as follows. 



Having formed a rectangle, whose sides AB and BC are 2 a 

 and 2 6; on these two sides as 

 diameters describe two semicircles, 

 and divide each of the semicir- 

 cumferences into equal parts^ 

 their numbers being as T : U; 

 from the points of section draw 

 lines parallel to the sides of the 

 rectangle, thus dividing it into a 

 multitude of minute spaces. Then, 

 beginning at the comer of any 

 one of these, trace a line to its 

 opposite corner, thence to the op- 

 posite corner of the next, and so on, until, having reached the ex- 

 2 



