412 BELL SYSTEM TECHNICAL JOURNAL 



The values of the Bessel zero, r, are not evenly spaced along the X axis; 

 indeed the density, or number per unit distance, increases as r increases. 

 Let the density be p{x). Then the problem becomes one of finding the 

 weight of a quadrant of material whose density varies as p{x). 



Suppose the expression for M, the number of zeros r, less than some value 

 X, is of the form 



M = Ax"^-]- Bx 



whence, by dififerentiation, 



p{x) = 2Ax-^B. 



The weight, IF, of the quadrant of a circle of radius R is then, by integra- 

 tion, 



W =\aR^ + ^ BR^. 

 3 4 



2L . . 2LW 



Since there are — lattice points per unit distance along the Y axis, 



ira iro. 



is apparently the total number of points in the quadrant. However, there 



are two small corrections to consider. First is that in this procedure a 



lattice point is represented by an area and for the points along the X axis 



Tra . . . . 



half the area, i.e., a strip — wide lying in the adjacent quadrant, has been 



omitted. Second is that the restriction w > for TE modes eliminates 

 half the points along the X axis. As it happens, these corrections just 

 cancel each other. Thus we have 



^ - 3 xr 2 X^ 



in which 



7 = ^^ S = ^aL Xo = ^ 



4 /o 



From a tabulations^ of the first 180 values of r, the empirical values A = 

 0.262, B = Q were obtained. This gives 



V 

 N = 4.39 -z . 

 Ao 



Subsequently, from an analysis of over a thousand modes in a "square 

 cylinder" (a = L), Dr. Alfredo Baiios, formerly of M.I.T. Radiation Lab- 

 oratory, has calculated the empirical formula 



N = 4.38 -3 + 0.089 ;-2 (2) 



Aq Aq 



