STRINGS 



The equation of motion of M is 



(2) 



where after the differentiations we must suppose x = I. This 

 gives 



nl nl I 



2c 2c = b' (3) 



where b is written for M/p, i.e. b is the length of string whose 

 mass would be equal to that of the attached particle. The 

 frequencies are therefore determined by 



where x 1) # 2 , x z , ... are the roots of the transcendental equation 



x tan x = l/b (5) 



Equations more or less of this character occur in many branches 

 of mathematical physics, and can often be solved approximately 

 by graphical construction. Thus in the present instance if we 

 trace the curves 



b 

 2/ = cot#, y= I ., (6) 



the abscissae give the roots. If 6 be relatively small these 

 fall a little short of JTT, |TT, |TT, ..., respectively, and the 



b-l 



Fig. 33. 



65 



