— = -^ sin 6 + zrTh cos0=—^k sin (n — 1)6 J n -i {kr) 



132 On the Passage of Electric Waves through Tubes. 

 As in (33), (34), 



^ = ^ COS 0_ ^ S m0=4*cos(n-l)0J„_i(fcr) 

 dx dr rdv 



-ikcos{n + l)6J n+l {kr), . . (43) 



dc dc . A , dc 

 dy dr r ad 



-iksm(n + l)6J n+1 (kr), . . (44) 



so that- by (19), (20) all the functions are readily expressed. 

 When n = Q, we have 



—^-kJ^k^cose, < ^ = -kJ 1 (kr)sind. . . (45) 

 a A ay 



For the circumferential and radial components of magnetiza- 

 tion we get 



z m dc 

 circ. comp. of mag. = b cos 6— a sin 6= -p- —775 



= -ij£-3 u (*r)mnn0, ■ ■ ( 46 ) 



1 r- n 7 • n * m ^ G 



rad. comp. or mag. =a cos 67 + sin 0= -jj -7- 



= "^ J n '{kr) cos n$, . . (47) 



corresponding to (37), (38) for vibrations of the first class. 



In like manner equations analogous to (39), (40) now give 

 the components of electromotive intensity. Thus 



circ. comp. = Q cos — P sin 0= j- 2 -7- = -f J n '.(kr) cosnO, (48) 



k dr /c 



rad. comp. = P cos 6 + Q sin 6 = — fv, — 77,= 7 v~ JJkr) sin n6. 

 r k-rd6 k 2 r v ; 



... .... (49) 



The smallest value of k admissible for vibrations of the second 

 class is of the series belonging to n = l, and is such that 

 kr' =■ 1*841, a smaller value than is admissible for any vibra- 

 tion of the first class. Accordingly no real wave of any kind 

 can be propagated along the cylinder for which p/V is less 

 than 1'841/r^ where / denotes the radius. The transition 

 case is the two-dimensional vibration for which 



c=^J 1 (l-841r//)GOS^ . . . . (50) 

 p = 1-841 Y//. . ^51) 



