SOME RESULTS OX CYLIXDRICAL CAVITY RESONATORS 423 



For case II, range limited by mode crossings, it is found that 

 A - .4o 



•To = 



F' = 



Bin'- - w'2) 



or - ■»/-)[»-/ - {n' + 1)'] 



Some values for this case are given in Table I. 



The formulas above are general and may be used for any pair of mode 

 types by using the appropriate values for A and /. 



The Elliptic Cylinder 



In the design of high Q circular cylinder cavity resonators operating in 

 the TE 01;/ mode, it is desirable to know how much ellipticity is tolerable, 

 so that suitable manufacturing limits may be set. The elliptical wave 

 guide has already been studied, notably by Brillouin^- and Chu,^^ but the 

 results are not in suitable form or of adequate precision for the present 

 purposes. More recently tables" have become available which permit the 

 calculation of some of the properties of the elliptical cylindrical resonator. 



The elliptical cavity involves Mathieu functions, which are considerably 

 more complicated than l^essel functions. ^^ The tables give the numerical 

 coefficients of series expansions, in terms of sines, cosines, and Bessel func- 

 tions, of the Mathieu functions up to the fourth order. These tables have 

 been used for the calculation of some quantities of interest in connection 

 with elliptical deformations of a circular cylinder in the TE 01« mode. 



The Ellipse 



All mathematical treatments of the ellipse (including the tables men- 

 tioned above) use the eccentricity, e, as the quantity describing the amount 

 of departure from the circular form. The eccentricity -is the ratio 



distance between foci 



e = . -. . 



major axis 



This is not a quantity subject to direct measurement, hence we here in- 

 troduce and use throughout the ellipticity, E, defined as 



_ difference between major and minor diameters 

 major diameter 



It is clear that the ellipticity is easily obtained directly. 



Again, many results are given in terms of the major diameter. Since we 

 are interested in deform.ations from circular, and in such deformations the 



