92 REPOKTS ON THE STATE OF SCIENCE. — 1916. 



Part II. 

 Bessel and Neumann Functions of Eqical Order and Argument. 



A small number of values of J„(a), J„_i(a), &c., are given in Meissel's 

 Tables^ of the J„(a) functions and the Committee's Tables of the G^{x) 

 and Y^{x) functions, viz. J„(a) from a = 1 to a = 24 and G„(a), &c., from 

 a = 1 to a = 13. The following tables have been calculated from the 

 formula- : — 



^■w=.v.3[©M^io(!Ni)-8xVo©M8) 



8100 \a/ V3/ 113400 Va/ V3/ 74844000 W \B/ J 

 For the G functions, 



-"(«)=UO^K^io©Mi)-^(!)MB) 



182300 \a/ \3) J 



«--(")=l[(!)M^e)MI)-s^©Mi)-.io©M|) 



8100\,a/ W 113400Va/ \3/ 74844000 \a/ \3j J 



The Y functions ^ are given by Y„(a;) = (log 2 — y)J^{x)—G^{x). 

 The numerical values occurring in the above formula are : — 

 log r(^) = 0-42796 27493 1426 : log r(f)-=0-13165 64916 8402. 



r(^)=2-67893 85347 077 : r(|) = l-35411 79394 264. 



The results were checked by means of the formula 



J»(«) Y„ . i(a;) - J„ . i(a;) ¥„(«) = -. 



A partial check was also obtained by the use of the Kapteyn Series 



»=- J^gs) _ --J^_i(2s^-1) ^1 

 f^, s-^ t, (2s-lp^ 2- 



The values of other functions of higher or lower orders are easily 



2n 

 calculated from the recurrence formula Z„_i(a;) Z„{x) + Z„^i{x) 



= 0, where Z„(a;) stands for Jn{x), G„(a;) or Y„{x). 



' and ^ Graj' and Mathews, Bessel Functions, pp. 266-279, p. 14. 

 ^ Phil. Mag. June 1916. 



