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LXXXVII. The Oscillations of Chains and their Relation to 

 Bessel and Neumann Functions. By John R. Airey, M.A., 

 B.Sc, late Scholar of St. John's College, Cambridge*. 



THE oscillations of chains afford interesting examples of 

 the practical applications of Bessel functions to physical 

 problems. These functions, in fact, first presented themselves 

 in connexion with the problem of the small oscillations of 

 a uniform chain suspended by one end — Bernoulli's problem. 

 The times of vibration in this case depend upon the roots of 

 the equation J (c) = 0. The more general function of the 

 same kind but of higher order, viz. J n (z), appears in the ex- 

 pression for the time of vibration of a chain whose line- 

 density varies as the nth power of the distance from the free 

 end. When a uniform chain is loaded at the free end — a 

 more general case than Bernoulli's, — the complete solution, 

 includes both kinds of Bessel functions, viz. J n (z) and Y n (^), 

 and their differential coefficients. The Y n (z) functions are 

 sometimes called Neumann functions -f. The following ex- 

 periments were carried out for the purpose of comparing 

 the observed periods of oscillation of certain " chains " with 

 those of "ideal chains" calculated from the expressions 

 giving the periods in terms of these functions. 



(A) Oscillations of a uniform chain. 



The periodic times r of the small " normal " oscillations 

 of a uniform chain of length l } suspended by one extremity 

 and hanging under the action of gravity, are determined by 

 the equation 



r=(4*/p)(llg)i, 



where p is a root of the equation J {z) = 0. The equation 

 J o (>) = has an infinite number of real positive roots corre- 

 sponding to the different modes of vibration of the chain. 

 The first root p x ?= 2*405 gives the period when the whole of 

 the chain lies on the same side of its original vertical position; 

 the second root /c> 2 = 5'520 gives the period when the chain 

 has one node; the third root p 3 = 8*654 gives the period when 

 the chain has two nodes and so on. 



In order to compare the calculated results with those 

 obtained by experiment, the times of oscillation of a long 

 chain were observed. A bicycle chain was employed so that 

 the vibrations might be restricted as far as possible to one 

 vertical plane. The observation of the periods presented no 



* Communicated "by the Author. 



t Gray and Mathews, ' Treatise on Bessel Functions/ p. 14. 



