168 V. OSCILLATIONS AND CYCLIC MOTIONS. 



so that the qp's are normal coordinates. Forming the differential 

 equations, 



38 



the integrals of which are 



Vl =A, cos 

 99) 



agreeing with the above result. 



The two normal vibrations are found, the first by putting (p 2 = 0, 



- in this case y = y 2 and the two beads swing together, the second 



normal vibration by g> = 0, f/ 2 = ~~ 2/i ? an( i the ^ wo beads swing in 



opposite directions with a frequency }/3 times as great as before. 



The middle point of the string is now at rest, or forms a node. 



The general case above treated is very interesting when we pass 

 to the limit as the number of beads is increased, giving us the case 

 of a continuous string, of the greatest importance in the theory of 

 musical instruments. 



Let us introduce in equation 94) the distance of the bead from 



one end of the string, 



rl . 



x = r a = ; - * 



Accordingly 94) becomes 



sn 



100) y (x) = x< P* s i n p cos (vs t a s }- 



3 = 1 



A glance at Fig. 37 shows us that, as we increase n, the ratios 

 at least of the smaller frequencies approach those of the integers, 

 1, 2, 3, .... By passage to the limit we may demonstrate that this 

 is exactly true for all the frequencies. 



If Q be the line density of matter of the continuous string, that 

 is, the mass per unit length, we have 



Accordingly since 

 we have in the limit 



Ql* 



Introducing this into the value of v tj 92), 



\c\-\\ 



^ 



