788 THE BELL SYSTEM TECHNICAL JOURNAL, JULY 1952 



Similarly the gradients of the J'-functions of the same variety as well as 

 the fluxes, are orthogonal, 



||(grad T(„))-(grad T(„)) dS = //(flux T(„))-(flux T(„)) dS 



(9) 

 = //(grad Tin]) • (grad Tim]) dS = //(flux T^) • (flux Ti^]) dS = 0, 



if m 5^ w. The following gradients and fluxes of the T-f unctions are ortho- 

 gonal for all m and 7i, 



//(grad T(„)).(flux T^m]) dS = //(grad ri„j)-(flux T(„)) dS 



= //(grad T(„))-(flux T^m)) dS = 0. 



(10) 



On the other hand, grad T[m] and flux T[n] are not, in general, orthogonal. 

 If all modes are present, the potential and stream functions are 



V = -VM(z)Tin){u,v), U = -IUz)Tm{u,v), ^ ^ 



(11) 



^ = -Vln]{z)TMiu, V), U = -lM(z)Tin](u, v) , 



where the tensor summation convention is used: whenever the same 

 letter subscript is used in a product, it should receive all values in a 

 given set and the resultmg products should be added. The negative 

 signs have been inserted m order to avoid a preponderance of negative 

 signs in later equations. Substituting in (9), we have 



Et = Vm grad T^ + V^] flux T^^] , 



(12) 

 Ht = — /(„) flux Tm + I[n] grad Tin] . 



The ^-functions for the various modes are determined by equation 

 (4) and the boundary conditions except for arbitrary factors related to 

 the power levels of the modes. If we choose these constants in such a 

 way that 



//(grad T) • (grad T) dS ^ x' ffr' dS = 1, (13) 



then the complex power carried by the wave is gi\'en by an expression 

 similar to that in an ordinary transmission line, 



P = WmII) + hVmiltn] . (14) 



