Bess el Functions of Zero and Unit Orders. 235 



can be expressed in the form 



dZ n n„ „ n 



ax x x 



— :. Z n + Z n _i — :Z„- Z„ + i, 



where Z„ is written for J n , G„, Y B , and other Bessel and 



Neumann functions. In the particular cases where n = 



and 1, 



dZo v n „ j ^Zi _ 7 Z x 

 - 1 — ! = — Zj and -— =A) . 



From these results, addition theorems of Z and Z x can 

 be obtained. They are usually employed in connexion with 

 tables of these functions where the intervals of the argument 

 are small — say, 0*1 or O01 ; but as the formulae can be 

 expressed in a simple and concise form, the calculation can 

 be carried out even when the increment or decrement is as- 



great as -~, which is approximately half the difference of two 



consecutive roots of the functions. Consequently, to evaluate 

 any one of the Cylinder Functions, of the first or second 

 kind, for values of the argument as far as 60, a short table 

 of the first 19 or 20 roots of the Z or Z x functions and the 

 corresponding values of Z 1 or Z for these arguments is 

 required. 



Bessel and Neumann Functions of Zero Order. 

 By Taylor's Theorem, 



Z (x + K) = Z <» + hZo'(w) + 1* Z "<» + . . . , 



and substituting for Z ', Z ", etc. their values in terms of Z„ 

 and Zj we find 



Zrf.+A) = |l- , +(U + - 4 (l-- 2 )-. . . JZ.(.) 



-{-£-f(-l)^(-i)--}^- 



If # is a root of Z (#) — say p, — the value of Z (p-\-h) 

 becomes 



z o( ,^)=-|_/ i -- 2TrgI (i^-) + -( r -) 





