234 Mr. Nicholson on Electrical Vibrations 



For these very short waves, the magnetic force is very 

 approximately 



£_ 



c = ~tt~ Ti . cos (k\/& + \ + e0 sinh (ky/b* + p — kb) . e lkvt (43) 



cc = /3 = 0, 



and is along the axis of.' the cylinders. 



The factor involving //, has been so adjusted that it cannot 

 become infinite at any point. 



In order to test the error involved in this approximation, 



we must find the asymptotic expansion under the assumption 



that n 2 is not zero, but a small positive quantity. 



1% 

 Let - = 0, where 6 is very small. Then 



^ + A-Wj-6 2 sinh 2 f)L = 0. 



If, following the previous method, we write 



we find 



0=| v^ + ^sinlr^^; . . . (44) 



«,■ 



-\lr — — = , as before. 

 y V <$>'' 



To the second order of small quantities, 



6" £ 



(j) = l> cosh f + ^r log tanh - ; 



and therefore j- s/b % + A is to be replaced by 



^ + £^$+x • • • (45) 



in the period equation. 



The ratio of the increase, to the original value, is of 

 magnitude 



loe 



b^b 2 + \ ' to 6-f- n/6 2 + A 

 Now # 2 =t^ while for light waves k is about 10 3 mm. 



Hence the correction is quite inappreciable, even when n 

 is fairly large. 



