50 



Mr. H. G. Savidge : Tables of the 



^i(#)> ^iWj etc. are the corresponding ker and kei forms; 

 thus 



X X (V) = ker 2 x + kei 2 x, etc. 



S(ar) = ber' x ker' x + bei' x kei' a?, 



T(a?) == bei' x ker' # — ber' x kei' a\ 



The series a, & a', £' are also tabulated. They are useful 

 for interpolation, and fi appears in Dr. Russell's formula 

 (107) (there called e). 



They may be defined by the equations : 



ber x = — cos /3, bei x = . sin 8, 

 V2irx V2ttx 

 and . — 



ker x = aJ^ e"' cos £', kei x = aJ ^ ^'sin £'* 

 where x is not less than 6, and 



j^ 1 25_ 13 



v/2 8\/2« 384 v^ 128^ 4 



p a _# 7T 1_ 1 25 



\/2 8 8^/2* Ifa»~"38V'20 8 



a'= — 1 25 _ 13 



tt ~ a/2 8\/2tf 38 V"2*' 3 128 ^ 4 



tf_ -JL-TL + Ji L + - 25 



P a/2 8^8V2^ 16^ 2 ^384^2^ 3 



When x is small, we have 



a = i log {27ra?X(#)}, yS = arctan (bei a?/ber <r), 



a' = -i-log < ^-^ r? /3' = arctan (kei a; /"ker a). 



Further Formulce (x greater than 5). 



Since ber x-\-ibeix = I (x\/i), we can find the values of 

 ber x and bei x by putting x\/t for x in 



t i <> e * f , . I 2 . l 2 -3 2 , l 2 ■ 3 2 . 5 2 , \ 



IoW ^7^ i 1+ s^ + \rwr + F(&? + -1 



(Gray & Matthews : BessePs Functions, p. 68.) 

 Making this substitution, noticing that 



v / ' = iKv / 2+ V^ + *^2- ^2}, 



