Bessel Functions of Zero and Unit Orders. 237 



Bessel and Neumann Functions of Unit Order. 



The Addition Theorem for the Cylinder functions Ji(#), 

 Y^A'), etc., can be found by expanding Z^x + h) by Taylor's 

 Theorem, and substituting the values of Zi(x). Z^'^x), etc., 

 in terms of Z and Z x ; it is thus easily shown that 



a , ( . +i) .[ 1 _^( 1 _i) + *(i_») + ...] iM 



4*-£-f('-5)*m( i -?)*-]«* 



If x is a root of Z x (x), say r, then 

 Z 1 (, + A)=[A-|-^(l-3) + ^(l-«) + ...]z oW 



r/, A If \ . , ¥ /, U , 25A 2 \ 



W A 3ft , 437. 2 \ 67i 9 /, 3A , \ 1 „ , , 

 -7T74 1 -^ + 2i^-'-) + 9!7K 1 -2-, + ---)--J Z ^' 



Therefore we have, approximately, 



Z,(r+A) = [ajsinA+A^ZoW, 



a] = 1_ 2(MT) and & = tM 1 + o>) • 



As in the case of the Z functions, for large values of 

 the argument or for small values of h, the following simple 

 formulae may be derived : — 



aDd Z x (r-h) _ (2r-h)(r+k) 



Z^r + A) (2r + A)(r-A)" 



Z (a?) and Z^r) could be calculated by means of the above 

 formulae to seven places of decimals for values of x greater 



than twelve. The following tables of ?log e (l+-l etc. 

 have been computed for values of - and - from 0*10 



