14 Roots ofBessel and Neumann Functions of High Order. 



these functions when the argument and order are nearly 

 -equal*. The formula (18^ holds tor the roots of Gr„(.r) 

 when n is large, if \ takes the value 



-p^' 



The Neumann function Y n (x) is equal to 



0og2- 7 )J,(*)-G.(*). 



Substituting (9) and (19) for J n (x) and Gr n (#), and proceeding 

 as before, we find that 



tan N tanfr-fr) ] = j4±g=|^ , . (21) 

 -We 2(lo g 2- 7) =() . 0738043 



7T 



Therefore 



n(tan^ 1 -</> 1 ) = 0-71172764-(/?-l)7r . . (22) 



to a first approximation, p — \ corresponding to the first root. 

 The first roots of Y n (a) are found as in the case of Q n (w). 

 For the second and higher roots, it is not necessary to go 

 beyond the second approximation from (22). Thus, using 

 five-figure logarithms, the first two approximations for the 

 second root of Y^a?) are 5*330 and 5*3549 f. The first five 

 roots of Gr 10 o(«£) and Y 100 (#) given below have been calculated 

 from (20) and (21), or by employing the Gr and Y functions 

 of nearly equal order and argument. 





Roots of Gioo(#). 



J- ioo(^')- 



1. 



104*380 



104*133 



2. 



112*486 



112*325 



3. 



118*744 



118*608 



4. 



124*275 



124-151 



5. 



129383 



129*266 



The approximation for the first root of Y 100 (#) from (22^ 

 gives ^=15° 44' 1"*5 and ^ = 103*90. 



Substituting this value of fa in (14) to calculate e, it is 

 found that the series in the numerator diverges from the 

 first, 15A z r' 6 being greater than A^. Consequently 103*90 is 

 the nearest value of the first root of Y 100 (#) by this method. 



* Phil. Mag. June 1916, p. 520. 



t Proc. Pbys. Soc. vol. xxiii. (1911). 



