172 V. OSCILLATIONS AND CYCLIC MOTIONS. 



Since for all values of t, y = for x = 0, we must have J. = 0, 

 and since y = for x = l, we must also have JE?sini'? = 0, that is 



116) vl = sit, 



where s is any integer, accordingly we obtain for the s th normal 

 vibration, 



117) v. - ?, 

 and the vibration is given by 



., ON -r, . Stt# (STtat \ 



118) y = ft sm -j- cos ( ^ a s j . 



The general solution is therefore represented as an infinite series of 

 normal vibrations, 



^ O N -n fSTtat 



103) y =2^ J5, sm cos 



s=l 



the arbitrary constants, B s , a s , being determined by the initial dis- 

 placements and velocities. In order to determine them let us make 

 use of the other fundamental property of normal coordinates, namely, 

 that the energy functions do not contain product terms. Let us write 



119) 



then 



120) 







I I 



= I <p{ 2 JV dx + | ri 







z 



o 

 Inasmuch as product terms in the gp"s are not to appear we must have 



z 







Putting X r equal to sin-^ this result is at once verified by inte- 



( 



gration. The property of normal functions expressed by equation 121) 

 is of fundamental importance in the theory of developments in infinite 



