CIRCULAR CYLINDER CAVITY RESONATOR 71 



At the first zero of Je (the value of x at a zero of j'i will be denoted by r), 

 4>l is held finite by choice of the value of p( . It is clear that (4) becomes 

 indeterminate at x = r, if 



Since // satisfies the differential equation 



j7 + -j(-h {1 - fyx')j( = (6) 



X 



and J(ir) = 0, one has by substitution 



Values of p for several cases are: 



^=1234 1 1 



n = 1.841 3.054 4.201 5.318 r-z = 5.331 r, = 8.536 



pf = 1.418 1.751 2.040 2.303 1.036 1.014 



4>iir)=-0.n6 -0.286 -0.446 -0.604 -0.180 -0.115 



Evaluation of 4>f{r) by the usual process- gives: 



Mr^ ^ -S^l^ (S) 



Values of (f)({r) are given in the preceding table. 



Since <p( is finite at the origin and at the first zero of Jf , it may be ex- 

 panded into a Maclaurin series whose radius of convergence does not, 

 however, exceed the value of x at the second zero of J( . Alternatively, 

 by choosing p{ to keep <^f finite at the second (or'^"") zero of J( it may be 

 expanded into a Taylor series about some point in the interval between 

 the first (or (k — 1)"') and third (or (k + l)"") zeros. Expansions about the 

 origin are given in Table I. 



Unfortunately, the convergence of these power series is so slow that they 

 are not very useful. Instead, equation (4) is used to calculate (l>( and 



/ 4>( dx is obtained by numerical integration. 



With pt fixed to hold 4>( finite at the first root, f i , of J( , it is soon found 

 that 4>f becomes infinite at the higher roots. This is because different values 



-Substitute (6) into (4) to eliminate JJ; dilTerentiate numerator and denominator 

 separately; use (6) to eliminate J^; allow x — > r, using J'Ar) = and value of p^ from (7). 



