STKINGS 85 



a point moving always with this local wave-velocity would take 

 to travel from the lower end to the point x, we have 



In terms of r as independent variable the equation (3) becomes 



0- ... ............... (5) 



For the present purpose we do not require the complete 

 solution, but only that solution which remains finite when 

 T = 0. This is 



where G is arbitrary, as may be verified by actual differentia- 

 tion, and substitution in (5). The function defined by the 

 series in brackets presents itself in many physical problems; 

 it is called the "Bessel's Function of Zero Order," and is 

 denoted by J Q (nr)*. Hence, inserting the time-factor, 



y = CJ (riT)cos(nt + 6) (7) 



The value of T corresponding to the upper end (x I) is 



T 1 = 2V(%), (8) 



and the condition that this end should be fixed gives 



Jo (nil) = (9) 



This determines the admissible values of n. The first few 

 roots are given by 



WTl /7r = -7655, 1-7571, 2-7546,..., (10) 



where the numbers tend to the form s J, s being integral. In 

 the modes after the first, the values of T corresponding to the 

 lower roots give the nodes. Thus in the second mode there is 

 a node at the point T/T, = -7655/1-7571, or a;/l = T*lT l * = '190. 

 The gravest period is 2?r/w = 5'225 \J(llg\ whereas the period 

 of oscillation of a rigid bar of the same length is 5130 ^(Ijg). 

 The comparison verifies a general principle referred to in 16, 



* Elaborate numerical tables of the Bessel's Functions, calculated by 

 Meissel and others, are given by Gray and Mathews, Treatise onBessel Functions, 

 London, 1895. A convenient abridgment is included in Dale's Five-Figure Tables 

 of Mathematical Functions, London, 1903. 



