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



however, that this does not interfere with the attainment of 

 our object. 



Putting for m the value ~ we get 



, sinh co (2 — e) sinh a> 



] cosh r 



cosh co (3 — 2e) cosh 3 co 

 (2- 6 ) (4-3e) sinho) 



(2-6) (4-36) smh*> 1 



+ (3 - 2e) (5 - 4e) cosh 5 a> + "^ J ' ^° ] 



which is valid except for co = 0. 



That this series is uniformly convergent for any range of 

 values of co greater than zero is shown if we write it in the 

 form 



{\A— p- + /^ 



-<0 . / x _ 1 



(3 — 2e) V cosh 2 co cos h 2 co 



(2-e)(4-3e) _ 



(3 - 2e) (5 - 4e) V cosh 2 co cosh 4 o> 



If now we select any real quantity 6 V greater than but 

 as nearly as we please equal to unity, we observe that the 

 successive terms of the above series are less than the corre- 

 sponding terms of the series 



{ l+ h + h ++ ) 



for ail values of cosh co >#i« 



Since the last is a convergent series of real positive terms, 

 it follows that our series is uniformly convergent through 

 any interval not including co = 0. 



We are therefore at liberty to integrate it term by term. 



Now we saw that 



Thus putting for y its value we have 



MRs-RQe 2 flf/ e £ C" da, x 



"= a(l-2e) Uj(l3i COsh "-j.— jr)A» 

 7 v cosh 1 ""*©' 



+ .-J cosh 1_= ?^rfG)+A 2 |, . ... (7) 



in which the meaning of the arbitrary constant k z has been 

 slightly altered for simplicity. 



