234 Dr. J. R. Airey on the Addition Theorem of the 



area of the passages. In the case of passages which are- 

 mere circular apertures of radii R and R' a simple result 

 maybe stated, for then Ci:c 2 = R:R' ; and, since p 1 2 =p 2 2 , 

 d : c 2 = S : S'. Accordingly 



X 2 /R'* 



-v/(!')S-(l)'. • • • M 



and the advantage of a small S' is even more pronounced 

 than in (13) . 



XXI. The Addition Theorem of the Bessel Functions of Zero 

 and Unit Orders. By John R. Airey, M.A., D.Sc* 



The Application of the Addition Theorem to the 

 Calculation of Bessel Functions of Zero and 

 Unit Orders. 



THE earliest form of the Addition Theorem of the J n (x) 

 functions was found by Bessel. In his notation, 



" welche Reihe zur Berechnung und Interpolation einer 

 Tafel dieser Functionen angewendet werden kann " f. 

 The expressions given by Lommel and others, 



J (z + h) = J (^)J (/0-2J 1 ( J )J 1 (/i)+2J 2 ( 2 )J 2 (/i)-..., 



J,(*4A) = J (*)Ji(A)-JiWJ>(*) + J.(«)J»CA)---- 



+j 1 ( z )j ih)-J 3 (z)j 1 (h)+ j&ys^h)-.. . , 



do not appear to have any useful application in the con- 

 struction of tables. 



A form of the Theorem, applicable over a wide range of 

 values of the argument, can be found in which one of the 

 terms in the argument is a root of a Bessel or Neumann 

 function of zero or unit order. 



The first differential coefficients of the functions satisfying 

 BessePs differential equation 



dx^x'dx^V xV V U ' 



* Communicated by the Author. 



t E. g., Meissel*s tables of J n (x) I ff=l, 2, 3, ... . 24. Gray and 

 Mathews, ' Bessel Functions,' pp. 266-279. 



