through Gases between Parallel Plates. 669 



Thus integrating term by term we get 

 1 



K- cosh 1-6 w I ; — Wco 



, , 1 f (2-e) 1 1 (2-e)(4-3<0 1 , , 1 ,o. 



Now we know that this integration is valid for all values 

 of cosh ft) > 1. 



If, however, we apply the usual test we find that the series 

 converges even when coshcw = l. 



Since the part of the series in brackets is a power series in 



r^ — , it follows by Abel's continuity theorem (see Chrystal's 



Algebra, vol. ii. p. 133) that the value of the series obtained 

 by putting cosh&) = l is continuous with those for which 

 cosh &) >1. Since these latter represent the integral required 

 and the integral itself is continuous up to cosh&)=l, it 

 follows that the above series (8) is valid for all real values 



of ft). 



We have yet a third integral to evaluate, namelv 



If -1 



— I cosh 1- 

 Putting as before cosh&> = ? this becomes 



codco. 



* J v/0 2 ^l 



Expanding this in descending powers of and integrating 

 term by term, we get : 



lf<9W0 _l-e 1_ f , , 1 1 , 1.3 _1_ I 



7 J \J&^\ " T~ 6l ' € l 1+ 2(26-1)0* + 2 a : 2 (4e-3# + + J 

 or 



jJcosh^*> dco = ^- 6 cosh^ft> { 1 + 2(2e-l)cosh 2 « 



+ 2^]2(4e-3)cosh 4 ft) + + J (9) 



The usual tests show that this series is convergent even 

 when cosh &> = 1. 



