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



trerae boundary, the line must be reflected from some one side 

 of the rectangle ABCD; the curve line thus indicated is the 

 trajectory. 



Did we attempt to find the algebraic equation of the curve by 

 eliminating t from its phoronomic equations, we would be ar- 

 rested at once by transcendental expressions ; yet on the assump- 

 tion of particular values for T and U, these disappear, and the 

 equations become finite. I shall examine only the two most 

 remarkable cases. 



First, let us suppose that the wire is equally flexible in all 



directions, in which case A = B, and since A. T^ = .B U^, 



T = U. The equations of the curve then become 



-art , TT (t — v) 



0? = a cos — ; y = o cos — ~- — - . 



If from these we eliminate t, we obtain, after all the reductions, 

 6« 07^ — 2 a 6 ^ 2/ cos ^^ + a2^2 ^ a^ ^^/gjj^ ^ y 



an equation at once recognised as belonging to the ellipse. The 



form of this curve depends mainly on the value of — ; we 



may therefore examine its most important varieties by attribut- 

 ing to V successive values from O to U. 



Wherfv is made zero, the equation takes the form hx-=.ay^ 

 which belongs to one diagonal of the containing rectangle. 

 When v\=. \ U, the equation becomes W x^ -\- a^ y^ z=: o? i*, 

 which is that of an ellipse whose axes are 2« and 26, and 

 which touches the four sides of the rectangle at their middle 

 points. Lastly, when sy = U, the equation is converted into 

 hx = — ay, which belongs to the other diagonal of the rect- 

 angle. If, then, a wire, which is equally flexible in all direc- 

 tions, be drawn aside, and then let go, its extremity will de- 

 scribe a straight line ; but, if any lateral impulse be communi- 

 cated to it, its extremity will describe an elhpse. 



Perfect equality in stiffness being unattainable, we have next 

 to inquire into the effects of a minute deviation from it. For 



this purpose, suppose that T = U + - , n being a very large 



n 



number. The trajectile setting out from the corner A, crosses 



