Addition Theorem oj Bessel Functions. 



1 





_ _3 

 ■ <-t ^4 + 32n" 



242 



and for Qo(#), when 



l_/«_l\l_/a 2 _3a\ 1 /17a 3 2 67a 9_\ J 



2 V2 S),r \4 8J/U 96 128/ ar 3 '" 



Similar expressions have been given for Pi(#) and Qi(#). 



When # = 9 and the calculation is limited to the con- 

 vergent parts of the asymptotic series, the values of J (9) 

 and Ji(9) can be found to about eight places of decimals. 

 The following table has been calculated by the above 

 method, and gives results correct to fourteen places : — 





x = 9. 



a- = 10. 



JoW • • • 



-0-09033 361118287 



-0-24593 57644 5134 



<*•(*) • • • 



. -0-39259 96475 9739 



-0-08744 806507746 



Y (*)... 



0-38212 71351 3807 



0-05893 63591 5000 



J 4 (*) . . . 



0-24531 17865 7332 



004347 27461 6886 



GX*)... 



. -0-16385 69515 5017 



-0-39115 25136 5956 



¥,(•) . . . 



0-19229 63187 7649 



0-39619 23750 1275 



Consequently the ascending series need only have been 

 employed for values of x from 001 to 8*00 to give Jo(#) 

 and Ji(#) to twelve places of decimals. 



The Zonal Harmonic P»(0) can be expressed in terms of 

 J (z) and Ji(s), where z — 6\/n{n+l). Lord Rayleigh's 

 formula, 



F n (0) = J (s) + 



12w(n + l) 



{z i J Q {z)-2zJ 1 {z)} 9 



has been extended and employed in the calculation of 

 P„(0) when n is large and 6 is a small angle ; the extended 

 formula, however, gives results correct to six places of 

 decimals even for comparatively large angles, e. g. when 



ir 



$=.- andn = 20. 



