520 Dr. J. R. Airey on Bessel and Neumann 



improvement of the experimental conditions more lines 

 would be observed. One additional point may be noted r 

 namely, that the interval between any consecutive pair of 

 observed lines is never less than that calculated from the 

 theory. 



Tt may fairly be claimed that the results brought forward 

 in this paper entirely support the theory of the relation 

 between infra-red and ultra-violet absorption advanced in 

 the previous papers, and there is little doubt of the validity 

 of the formula 



13951-6 +p x 223*225 + nX 2'73 



ifor the less refrangible ultra-violet absorption band of sulphur 

 dioxide. The whole of this band group therefore has accu- 

 rately been calculated from the infra-red absorption spectra 

 of sulphur dioxide and hydrogen sulphide, the only additional 

 fact being used of the central wave-number of the ultra- 

 violet band which enabled the true wave-number of the 

 fundamental infra-red sulphur dioxide band at 7*4:07 fj, to b^ 

 determined. ' 



The University, Liverpool. 



LXIII. Bessel and .Neumann Functions of Equal Order and 

 Argument. By John R. Airey, M.A., D.Sc* 



TI^HE Bessel and Neumann functions J n (s), Gr„(~), and 

 JL Y n (z) of equal or nearly equal order and argument 

 occur in many physical problems, and various authors have- 

 given formulae expressing their behaviour under these 

 conditions. 



Graf t found that when n is very large, 



Further and more general formulae t, to order -, were 

 tt n 



found from the relationship between the Bessel and Neumann 

 functions when the argument and order are nearly equal^ 



* Coram uuicated by the Author. 



t Graf u. Uubler, ' Theorie der Besselschen Fimktionen,' Erstes HeM, 

 96-104. 



X Nicholson, Phil. Mag. Aug. 1908, pp. 276-279. 



