336 BELL SYSTEM TECHNICAL JOURNAL 



We shall use the relations* 



VaC tanh T aC = k~^ [yc tanh yc]d k 



(5.1-7) 

 TaC cosh TaC = K~'^ [yc coth yc]d k 



where the subscript d on the brackets stands for "diagonal" matrix, the 

 i^^ element in the principal diagonal of [yc tanh yc] a being 7»(; tanh 7iC where 



Ti c = c^T/a (5.1-8) 



and \i is the ith latent root of a F^ . In our applications yi is either posi- 

 tive real or positive imaginary. 



From (5.1-6) the Xi's are the roots of 



X + 9.086 1.785 5.178 



-.157 X - 17.218 19.362 



-.996 13.566 X - 38.329 



(5.1-9) 



= X3 - 46.461 X2 - 101. 96X + 3464.5 = 



and have the values 



Xi = -8.886, X2 = 8.284, X3 = 47.06 (5.1-10) 



The elements K21 , /C31 of the modal row [1, K21 , /C31] corresponding to Xi 

 may be obtained by solving the two equations derived from the last two 

 elements of 



[1, K21 , KsiKXi/ - aV„) = (5.1-11) 



namely, 



1.785 + (Xi - 17.218) K21 + 13.566 K31 = 



5.178 + 19.362 K21 + (Xi - 38.329) kzi = 



When the value of Xi from (5.1-10) is used these equations yield 



/C21 = .1593, /C31 = .1750 



Likewise, the first and third elements of 



lKn,l,Kz2]{\2l - aVl) = 



and the first and second elements of 



[/C13 , K23 , 1](X3/ - arl) = 



* This is the modal row matrix analogue of equation (11) in Section 3.6 of the Refer- 

 ence*. The modal rows of Ta are equal to the modal rows of a^Pa. 



