The Oscillations of Chains. 



737 



difficulty when the chain was vibrating in the first and second 

 modes, but when the chain had three or four nodes, only a 

 limited number o£ vibrations were executed without assistance, 

 and it was necessary in these cases to maintain the motion 

 by gentle pressure of the hand near the top of the chain. 

 The error thus introduced is however quite small. 



Twenty sets of 100 vibrations each were recorded for each 

 mode of oscillation of the chain. The time was measured 

 by means of an accurate stop-watch. 



This experiment, which is quite easily performed, is perhaps 

 the simplest example of a physical problem involving the use 

 of Bessel functions. 



Value of g at Morley, Yorks = 981*4 cms./sec. 2 

 Length of chain = 219*9 cms. 



Mode of vibration. 



Time of vibration 



in sees. 



Calculated. 



Time of vibration 



in sees. 



Observed. 



First (no node).. 



2-473 



2-470 



Second (one node) . . . 



1-077 



1-075 



Third (two nodes) ... 



•687 



•685 



Fourth (three nodes) . 



•504 



•504 



Fifth (four nodes) . . . 



•398 



•397 



(B) Oscillations of a heterogeneous chain ivhose line-density 

 varies as the nth power of the distance from the free 

 end. 



This extension of Bernoulli's problem is due to Prof. Sir 

 Geo. Greenhill. The form of this chain, when executing its 

 principal oscillations, is given by 



n 



y = Ax 2 J n (2b^/x) sin (pct + k), 



where 4c 2 — g, 4:b 2 =p 2 (n + 1) ; x is measured upwards from 

 the free end, and y is measured horizontally. The fact that 

 the upper end is fixed imposes the condition that 



J n (2b yT) = 0. 



If p he one of the roots of this equation, / the length of 

 the chain, and t the time of vibration, it is easily shown that 



r=(47r/p)[(n + l)^]i {1) 



