240 Dr. J . It. Airey on the Addition Theorem of the 



and in the Report of the Mathematical Tables Committee of 

 the British Association (1916) to eleven places for A = 0,001 

 to 1-600. 



The first forty roots of Jo(^) and the corresponding 

 values of Ji(a?) were published by Willson and Peirce *,. 

 J](^) being given to eight places of decimals (see p. 241). 

 This table, in conjunction with those given above, can 

 therefore be employed in calculating J (#) for any value of x- 

 between 15 "0 and 126*0 to six places of decimals. 



The first fifty roots of Ji(#) and the maximum and 

 minimum values of Jo(#) have been calculated by Meissel \ 

 to sixteen places of decimals [see p. 241). Ji(x) can therefore 

 be found for values of x from 15*0 to 159'0. 



These tables, to four places only, are given in Jahnke u. 

 Emde's Funktiontqfeln. 



The most complete tables of Jo(#) and Ji(#) are those 

 calculated by Meissel J from Ihe ascending series to twelve 

 places of decimals: # = 0'00 to 15'50 by intervals of 0'01 : 

 for larger values of x, the asymptotic series can be employed 

 where the calculation is not carried beyond the least term 

 of T (x), Q (x), etc. 



J (a) = V ^[ P oW c <> s (-»-j)-QoW sin (^ _ f)J 

 and 



3\(x) = V ^[ p iW sin^-^ + Q^cos^-^J. 



From a consideration of the divergent part of these series,, 

 it has been shown § that a greater degree of accuracy can be 

 obtained by resolving these into series which can be evaluated 

 by Euler's method of summation. In this way it is found 

 that the divergent part of an asymptotic series of the first 

 kind where the signs of the terms alternate, is equivalent to 

 the least term multiplied by a "converging factor/ 7 In 

 these cases the term independent of x is J. # 



When x is an integer n> the "converging factors" for 

 P Q (x) and Q {x) are : 



111 15 103 



+ o~~9 -t ."> u .. a ~r " 



2 Sn^Sn 2 USn* ' 1024n 4 * * 



«"* 1,1 .J>_ ?_ , 159 



2 biiSn 2 12tm 3 " 1 ~1024n 4 °" 5 



* Willson and Peirce, 'Bulletin of the American Mathematical 

 Society/ vol. iii. 1896-97, pp. 153-5. 



t Gray and Mathews, ' Bessel Functions/ p. 280. 



X Gray and Mathews, 'Bessel Functions/ pp. 247-266. 



§ Archiv der Math. u. Phys. 1914. 



