nates passing tlirough O, where the y-axis touches the parabohi, 

 the form of the string is represented by the equation: 



y^ = 2px. 

 The potential energy of any small particle dy of tlie string 

 amounts to : 



^-HIQRdy. 



Now QR -=11 — X in which A represents the maximal deflection 

 OC of the string, whilst x is the abscissa of the point Q. As 



.r=i" — we obtain tor the potential energy ot the part dy : 



2^ o. 1 ^ , 



2 V 2^; 



dy. 



By integrating this expression between the limits and -/, 



I being the length of the string, we get half the potential energy. 

 Hence the total energy amounts to : 



^ipot) = 2 X ^J T/^ - |;Y dy z= i IhHI . 







In order to calculate the kinetical energy, each part of the string 

 is assumed to perform a series of damped oscillations, which may be 

 represented by the expression : 



in which S be the place of the part at any moment with regard 

 to the line BD, A the maximal amplitude, a the damping-con- 

 stant, o) the number of oscillations in In seconds. 



If we consider a particle of the string dy situated in Q. we have : 



Az=QR = h — 

 The velocit}^ at any moment is : 



On 



V 



dt 



I A I jo) sm oit -\- a cos o)t] E'-'-^'- 



which expression at the moment when the string passes through its 

 tinal position of rest, becomes : 



Vmnx. = — (A — -)ü>e 



Hence in the formula for the kinetical energy 



to 



