SOME RESULTS ON CYLINDRICAL CAVITY RESONATORS 417 



To show that any value of ^ > reduces P below its value when ^ = 0, 

 let 



a = cos^ (p -{• - sin^ <p 

 P 



b = { cos (f — - sin ^ 1 sin^ ip 

 c = {l/r)\ 



It suffices to show that 



a a -\- he 



where the question is in doubt because h may take on negative values. If 

 the inequality is to be valid, it is necessary only that (i + a) > 0, that is, 

 cos «^ > 0. Hence, for the TE modes, only I — ^ needs be considered. For 

 this case, the expression for P simplifies to 



r 1 



P = 



For the TM modes, there is similarly obtained 



27r 3 , 1 . 3 ' (4') 



cos ^ + - sm (^ 



P = 



P = 



r 



1 



2-K , 1 . w > (9) 



cos V? + - sm ip 



r cos (p 



2-K , 1 . « = 0. 10) 



cos v? + ;r- sm <p 



Ip 



It is easy to show, since cos ^ < 1 and sin ^ < 1, that both (9) and (10) 

 are less than (4'). 



Hence we have shown that, under comparable conditions, i.e., r and p 

 constant, the TE Omn modes have higher values of P than any others. 

 There is one flaw in the argument, viz., r takes on discrete values and cannot 

 be made the same for all modes. It is conceivable, therefore, that for some 

 specific values of P, a mode other than the TE Omn can be found which 

 gives a smaller W than either of the two "adjacent" TE Omn modes, one 

 having a value of r higher, the other lower, than the supposed high-P 

 mode. This situation requires further refinement, and hence complication, 

 in the analysis; we pass over this point. 



Having so far indicated that the TE Omn modes are the best, our next 

 objective is find the best value of m, if possible. 



