and Inductance of a Concentric Main. 549 



Similarly in the outer conductor we find that 



g"W«"" /a_yfl rcosh m(c-r) x / : 2 4- cos m ^-r^r i 1 / 2 z^ + m 

 «o ~~ y/2irmr \b J Lcosh ??i(c — 5)^/2 — cos ??i(c — />) v /2-l 



. '. . (in) 



where 

 and 



\c-b)^/ 



—m(c—r)/ Vi 



tan0 = 



sin 6t + e sin e 5 



—m(e—r)/ si 2 m{c-r). \! 2 ' 



e cos e, + e cos e 2 



7"2 + » 



£2=S+ 7-2 + 8 ; 



tan 7= -r ; tan 8= — and y — S = ??ic \/2. 



Since {coshm(c — r) \/2+ cos m(c — r) \/%}fr 



continually diminishes as r increases, we see that the ampli- 

 tude of! i always diminishes as r increases. 



VI. Concentric Main with Hollow Inner Conductor. 



Let us suppose that a x is the inner radius of the inner 

 conductor. The solution (58) for the current-density i still 

 applies. Assuming that i = ? cos> cot, when r is a l5 we get two 

 equations connecting the four constants A, B, C, and D. 

 The equation corresponding to ((52) is now 



~d'i \tt\x 



p ¥ - 



r }.,. nt 



(112) 



Substituting the value of i in this equation and equating 

 the coefficients of cos cot and sin cot in the resulting equation 

 to zero, we get two other equations connecting the four 

 constants, and hence they can be found. 



Adopting the method suggested by 0. Heaviside*, however, 

 the values of the effective resistance of each tube may be 

 written down at once from the formulas given above. From 

 formula (101), for instance, the resistance R of the outer 

 conductor is given by 



K »-^-6»)L 1+m V 192 " + 8( C 3 -&y 0g 6 



-icS^K) 2 }]- • • ^ 



* ' Electrical Paper?/ vol. ii. p. 192. 



