Bessel Functions of Equal Order and Argument. 237 



third and following terms, exceedingly intricate algebra 

 would be necessary. 



(IV.) The explanation of the fact, mentioned by Dr. Airey 

 on p. 528, that all the formulas for J n (n) etc. are true when 

 n is not an integer will be seen from (III.) above ; for these 

 formulas really have to be derived from the Bessel-Schliifli 

 integral, or some such result, which holds whether n is an 

 integer or not. 



(V.) It would have added considerably to the interest of 

 Dr. Airey's paper if he had given some indication as to how 

 formula (16) on p. 52G for J- n (n) is obtained without making- 

 use of the methods employed by Debye. 



I am, Gentlemen, 



Yours faithfully, 



Trinity College, Cambridge. G. X. WaTSON. 



June 17th. 1916. 



XXVI. Bessel Functions of Equal Order and Argument. 



To the Editors of the Philosophical Magazine. 

 Gentlemen, — 



AS Prof. Watson surmises, the object of my paper was 

 to obtain formulas for the calculation of J w (n), Y„(?i), 

 etc. for integral values of the order and argument and was 

 not intended in any way as a contribution to the theory of 

 these functions. 



Formulas for J w (?i), J_„(?z), Y n (n), etc. had already been 

 given, and for the purposes of the paper, the calculation of 

 further terms was all that was required. I can corroborate 

 Prof. Watson's statement that intricate algebra is necessary 

 for the computation of the third and following terms. 

 However, Bessel's Integral appeared to afford a simple 

 method of deriving the asymptotic expansion of J n {x) in 

 the case under consideration. Although divergent, integrals 

 were employed and errors of small order introduced thereby, 

 it is interesting to note that, owing to these errors cancelling 

 one another, results were obtained which were in agreement 

 with those derived from the Bessel-Schlafli integral and the 

 contour integrals of Sommerfeld and Schlomilch. 



From the Bessel-Schlafli integral, Prof. Nicholson found 



that, to order -, J n (&) can be equated to the asymptotic 



expansion of Airy's integral as in formula (2). Debye's 

 formulas for J„(#) and J_»(#) derived from Sommerfeld's 



