672 BELL SYSTEM TECHNICAL JOURNAL 



7 d?+fd?^ '''-'' = -J-d¥^^~' (12^) 



both members of the equation being, by the famihar reasoning, equal 



to a constant which I denote by X-. It follows that the function F{d) 



is of the form: 



F{d) = CcosXe -]- D sin Xd (126) 



and the coefficient X thus far seems to be unrestricted. But it carries 



its own restrictions in itself; for the coordinate is a cyclic coordinate, 



like longitude on the earth; whenever it is altered by lir, we are back 



at the same place. The function F(d) must therefore repeat itself 



whenever 6 is altered by 2x; but this will not occur, unless X is an 



integer : 



X = 0, 1, 2, 3 •••. (126a) 



These are the Eigenwerte, and the functions (126) with one or another 

 of these values assigned to X are the EigenfunkHonen, of the equation 

 (125). In this case we have obtained Eigenwerte for the parameter 

 and EigenfunkHonen for the solutions of a differential equation, not 

 out of boundary conditions but out of the simple fact that the inde- 

 pendent variable is by its nature cyclic. Such cases will occur in the 

 undulatory mechanics. 



We arrive at the third and last step of the problem : the determina- 

 tion of the function /(r). It is governed by the differential equations: 



a distinct equation for each of the permitted integer values of X. As 

 the solution of such an equation as (115) is a sine-function of the 

 variable mx, so the solution of such an equation as (127) is a function 

 of the variable mx; not however a sine-function, but a Bessel function. 

 For the values 0, 1, 2, • • • of X, the solutions of (127) are the Bessel 

 functions of order 0, 1, 2, • • •, denoted by Jo{mr), Ji{mr), Mmr), and 

 so forth. 



Like the sine-function of mx, the Bessel functions of mr oscillate 

 back and forth between negative and positive values as their variable 

 increases from zero to infinity, and pass through zero at an infinite 

 number of discrete values of mr. These do not lie at equal intervals, 

 as do the values of mx at which sin mx vanishes. Their values may 

 be found in the tables; I shall designate them as b^, b~, 6^ ••• in 

 order of increasing magnitude, using the superscripts not as expo- 



