1154 THE BELL SYSTEM TECHNICAL JOURNAL, NOVEMBER 1952 



other hand, tlie odd modes of this family are all given approximately 

 by 



tan xs = 0, (398) 



since the hji^erbolic cotangent on the left side of (387) will never be 

 small for the same value of F^ (or x) as the hj^jerbolic tangent. 



Equation (397) has an infinite number of positive real roots, which 

 may be denoted by 



Xi , X3 , X5 , • • • , X2p+i , • • • , (399) 



and it is clear that 



pr/s < X2P+1 ^ (p + Dtt/s. (400) 



If the thickness h of the main dielectric is not zero, so that the right 

 side of equation (397) is finite, the higher roots X2p+i approach nearer 

 and nearer to pir/s as p increases; but if 5 = 0, then 



X2P+1 = (p + ^)7r/.s = (2p + l)T/a (401) 



for all p. The positive roots of equation (398) may be called 



X2 , X4 , X6 , • • • , X2p , • • • , (402) 



where 



X2p = pyr/s (403) 



for all p; and both sets of roots may be combined in the single sequence 



Xi , X2 , X3 , • • • , Xp , • • • . (404) 



The advantages of designating the principal mode as the first rather 

 than the zero-th mode seem to outweigh the minor disadvantage that 

 in the sequence (404) the odd subscripts correspond to what we have 

 been calling the even modes, and vice versa. 



The attenuation and phase constants of the pth. mode are obtained 

 in terms of Xp from (388) and (396). Under the usual assumption that 

 the attenuation per radian is small, we have 



