206 BELL SYSTEM TECHNICAL JOURNAL 



parts of well-known series that define certain elementary functions. 

 It can be verified readily that 



U = cos T -\- '— 



F=sin.+,[log(l+^)-^]+^ 



(46) 



+ TT7fl+ •••. (47) 



i2oy 2403; ' 24oy 



U' = — sin T + ^ 1- -— (above remainder of (3)), (48) 



T 



V d 



V = COS T 1- T- (above remainder of (4)). (49) 



1+- ^' 



y 



The series (46) to (49) possess a certain advantage for computing in 

 that the quantities in brackets are real numbers. ( Note that 



- = ^^-7 • \ They have been used also in obtaining the approxi- 

 mate formulas given in the body of the paper. 



The quantities discussed above all pertain to the sheath. For 

 finding Z//, involved in the last of the formulas (9), the series are not 

 valid, of course. For this we have the well-knowii series, 



1 /o(.rO 1 



U,' Jo'{xO .Vi 



^4 ^96 1536 ^ 23040 



(50) 



— see, e.g.. Gray, Mathews and McRoberts, " Bessel Functions,' 

 2d edition, p. 170. 



