670 Mr. A. A. Robb on the Conduction of Electricity 



It thus appears that the three series given by equations 

 (5), (8), and (9) are valid expansions for all real values of co, 

 and therefore may be used to calculate tables of the required 

 functions. 



It will be shown later that co = corresponds to the position 

 of one of the plates. Thus the values of the functions for 

 <w = are of special importance. Although the series which 

 we give are convergent for this value of &>, yet the conver- 

 gence is very slow, and it appears desirable to obtain other 

 expressions for them in this case. 



Let us first take 



- — cosh 1 "' 



1 — 6 



For a) = this becomes 



1-e 



•Jo 



C dco 



\ " H 



J cosh 1_e 



J_ 

 cosh 1- 



Putting cosheo=£>-£ we have 



6 If 1 _J . 



and therefore 







dco el 





L 1-62 



,—*-• - -\2(l-e) 



rfeNK) 



F \2(l^) + V 



This may be at once evaluated by means of the table of 

 r functions. 



We shall next find the value of the series given by 

 equation (8), when co = 0. 



The series in this case becomes 



^?(M , 1 (2-e)(4-3e) , 1 (2-e)(4-3e)(6-5 e) -, . 1 . 

 2 I (3-2e) + 2 (3-2e)(5-4e) + 3(3-2e) (5-46) (7-6e) + + J (11) 



Consider now the general hypergeometric series of argu- 

 ment unity F(a, b, c, 1). This is convergent if c—a—b is 

 positive. It has then (see Whittaker's ' Modern Analysis/ 

 p. 241) the value 



T(c)T(c-a-b) 



rcc-bKr(v-ay 



