200 Dr. J. B. Airey on Bessel Functions 



this he had been anticipated by Young. It is desirable in 

 the interests of clearness, that the work of Newton and 

 Helmholtz should not be confused and the name trichromatic 

 theory not employed, when the Young-Helmholtz theory is 

 referred to. 



We are indebted to Mr. Alex. H. Gray for assistance with 

 some preliminary work involved in the investigation, and 

 to Prof. Karl Pearson for two letters of advice and criticism, 

 which were o£ great help in planning the investigation. We 

 are also indebted to Sir Oliver Lodge for criticism, which 

 enabled us to effect a considerable improvement in the latter 

 part of the paper. 



XVI. Bessel Functions of small Fractional Order and tlieir 

 application to problems of Elastic Stability. By John R. 

 Airey, M.A.,D.Sc* 



MANY problems of elastic stability depend for their 

 solution upon the roots of Bessel functions of small 

 fractional order, e. g. the stability of a flat bar .or blade 

 bent in its own plane f, or of a long thin rod placed 

 vertically and clamped at the lower end %. For a deep 

 horizontal beam fixed at one end and supporting a load at 

 the other, the condition of equilibrium is given by the roots 

 of J_i(a')=0 ; and for a beam carrying a uniform load, the 

 roots of J_i(a?)=0. Generally, if the bending-moment is 

 proportional to sc n , where x is the distance from the 1'ree end 

 of the beam, the condition of stability is found from the 



1 / z \ 2p 



roots of J_p( / s) = 0, where p= 777 tt an d I rr- I =>. 



v J l 2(n+.l) yip) 



When the clamped vertical rod is of uniform circular 



section, the height consistent with stability is subject to the 



condition that J_i(a?)=0, and in the general case, where 



the rod is a solid of revolution, if r the section radius at a 



depth x below the top =X# m , and W the weight of the rod 



above the section = fix 7 \ the solution of the problem depends 



upon the roots of J p (a?) = where p— -. 9 - This 



* Communicated by the Author. 



t " Elastic Stability of Long Beams under transverse forces." A. G. 

 M. Michell. Phil. Mag- 5th series, vol. xlviii. Sept. 1899, pp. -298-809. 



" The Buckling of Deep Beams." J. Prescott, Phil. Mag. 6th series, 

 vol. xxxvi. Oct. 1918, pp. 297-314. 



t "Height consistent with Stability." Sir George Greenhill. Proc. 

 Camb. Phil. Soc. vol. iv. Feb. 1881, pp. 65-73. 



