232 Prof. G. N. Watson on Bessel Functions of 



or, in terms of the wave-length X, 



8tt y/i dx 



x" / */ 



(**-i) 



(8) 



in agreement with Planck's law. 



In the second assumption above the essential requirement 

 is the existence of a limiting energy, hf in each mode. For 

 this it is not necessary that emission of energy should take 

 place suddenly when the limiting energy is reached. Emission 

 and absorption may be continuous processes, and the attain- 

 ment of the limiting energy condition may involve some 

 other action than that postulated, such as the disruption of 

 the molecular system with possibly very little transference 

 of energy to the aether. 



XXy. Bessel Functions of Equal Order and Argument. By 



G. IS. Watson, 31. A., Assistant Professor of Pure 

 Mathematics at University College, London. 



To the Editors of the Philosophical Magazine. 

 Gentlemen, — 



MY attention has been directed to a paper under the title 

 " Bessel and Neumann Functions of Equal Order 

 and Argument" by Dr. J. K, Airey (Phil. Mag. June 1916. 

 pp. 520-528). The ultimate object of that paper is to 

 obtain a number of numerical results concerning Bessel 

 functions ; with these results I am not at present concerned. 

 But in the theoretical work which Dr. Airey gives at the 

 beginning of his paper, on which his numerical results are 

 based, there are a number of statements for which more 

 complete investigation appears to be necessary. Some of 

 these statements contain errors which seem to me to be 

 serious from the theoretical point of view, while others are 

 merely liable to be misinterpreted by the casual reader. 



Before enumerating the statements to which exception 

 may be taken, I wish to emphasize the fact that these errors- 

 do not, so far as I can see, affect the numerical results con- 

 tained in the paper. 



(I.) Graf's approximation for J n (n) is incorrectly given 

 (p. 520) as 



the correct formula is 



but this is obviously an oversight. 



