TKAJiSACTlOAS OF SECTION A. 595 



FHIDAY, SEPTEJUBER 4. 

 Joint Didcussion with Sections B and G on Gaseous Exjilosions. 



Department op Mathematics. 

 The following Report and Papers were read : — 



1. Beijort on Bessel Functions. — See Reports, p. 58. 



2. The Asyinj)totic Expansions of Bessel Functions. 

 By J. W. Nicholson, D.Sc, B.A. 



The tabulation of Bessel functions, so far as it has yet been carrit d out, has 

 been chiefly based upon the formulae 



J(.) = ^-C0s(z + a--^--^J 



4/r-r-.4>/ --3- 

 ^ -^- 2 ! (8sf + • • . 



^- 8z 3 ! {8zy + • • • 



as, for example, in the case of a tabulation now being made by a Committee of 

 the Association.^ 



These formulae are only useful when s is fairly large and n small in com- 

 parison. Suitable bases of calculation for larger values of n have recently been 

 obtained by the author. There are three distinct cases for the Bessel functions of 

 real argument. 



Case 1 : n less than s. 



^^^ sin p 



— cos p (m not an integer) 



ttz 



Y«(2)= — */——- COS p (ti an integer) 

 ■where Y„(z) is the solution defined by 



Y„(.) = {9^;^-(-)..3Jg(!))(„ = i.teger) 

 Then if n = z sin a, and a is not too close to 90° 



R = sec a + ;> sec^ a + - ■* sec* a + . . . 



where 4 (s + 8) X,^.3 + (s + 2)^ X,+i + 2n~s . s + 1 . s f 2 . X._j + n*s , s---i . X,_, = 



\ - ^ \ _ 27 -96m- . _4640rt-^-1125-640w* 

 '^i - 23' * 2' ' " ^2'" 



and the second, fifth, eighth, &c., terms of E, are two orders in n or z less than 

 those preceding. 



' £.A. Itejm-t. 1007, p. 94. 



QQ2 



