I 



through Gases between Parallel Plates. 667 



By successive applications of this formula we get 



* dco 1 sinh co 



cosh m co ~ m cosh" 1+1 co 



(m + 1) sinh co (m + l)(m + 3) sinh co 



*s at 



m(m + 2) cosh TO+3 co m{m + 2) (m + 4) cosh™* 6 w 

 _ (m-H)(m+H) ,.,(m + 2r — 1) P w dco 

 mim + 2)(m + 4) . . . (?7i + 2r — 2 ) J ^ cosh w+2r co ' 



The former method of treatment shows, however, that 

 dco 



cosh m + 2r co 



1 ( 1 (m + 2r) | 



(m + 2r) cosh m+2r a) C. 2 (m + 2r + 2) cosh 2 co J 



Thus the remainder is 



(m + 1) Q + 3) (m + 2r — 1) 1 



~~ m(m + 2) (m + 4) (m + 2r) cosh w+2r o> 



f , 1 ("*+2r) 1 



L 2 O + 2r + 2) cosh 2 co + 

 The coefficient 



I' 



_ (m + l)(m + 3) (m + 2r~l) 



m(m + 2) (m + 4) (w + 2r) 



is clearly a proper fraction which decreases in absolute value 

 as r increases. 

 The expression 



1 j 1 (m + 2r) \ 



cosh m+2r co \ 2 (m + 2r + 2) cosh 2 co '" ) 



clearly has the limit zero so long as cosh co > 1. 

 Thus the remainder vanishes for ?* = cc . 

 We can therefore write 



P w dco 1 sinh co (?n + 1) sinh co 



J ^ cosh" 1 &) — m cosh 7 " 4 " 1 co m (m + 2) cosh TO+1 co 



and this is valid so long as cosh co>l. 



If, how r ever, cosh co = 1 then sinh co = and each term of the 



series vanishes. Since we know that the integral l — -. 



° J^cosh^co 



is not zero, it follows that the function represented by the 



series has a discontinuity for co = 0. It will be found. 



" 2Z2 



