Neumann Functions of High Order. II 



successive approximations for the roots of J 10 o(<v) are as 

 follows : 



No. of root. 



*. 



p P . 



€. 



P =i. 



23° 9'42"-5 



108-767 



0-02755 





23° 14' 51"*7 



108-837 



0-02726 





23° 14' 48"-5 



108-836 





p=2. 



30° 12' 32"*5 



115-715 



0-01250 





30° 13' 48"-5 



115-740 





p = 3. 



34° 39' 3"-4 



121-561 



0-00806 





34° 39' 38"-2 



121-575 





p=i. 



37° 58' 35"-l 



126-861 



0-00596 





37° 58' 55"*3 



126-871 





p = 5. 



40° 39' 23"-4 



131-817 



000473 





40° 39' 36"-5 



131-824 





Roots of Jio(ff), J] 



ooO'O* and JioooW- 



Roots of Jio(ar). 



Jioo(#). 



Jiooo(#). 



1. 



14-4755 



108-836 



1018-62 



2. 



18-4335 



115-740 



1032-76 



3. 



22-0470 



121-575 



1044-39 



4. 



25-5095 



126-871 



1054-74 



5. 



28SS63 



131-824 



1064-24 



For very large roots of J„(#), fa is nearly equal to 90° 

 and its value is not easily determined from (11). Using 

 seven-figure tables, any root can be found to six significant 

 figures when fa is not greater than 85°. 



If fa exceeds this value, the sine of its complement ± can 

 be calculated from the formula 



. k 3 13* 5 ^_ 61« 7 /iK . 



sin <?!=«+ ^ + — + _..., . . . (15) 



wh 



ere 



a: = 



20 ' 80 



1_ An 



\ - 7r(4p-l + 2»)' 



a ^ x 2 19 

 cosec^X-^-^-^..., . . (16) 



and ppsrncosec^. 



A more accurate value of cosec : can be obtained from 



1 2 19 „i,N 



where _ _i_ , /_L _^L \ I 



/ A - A,+ 8n 8 X + V3n 2 384w 4 / '\ 3 '" 



1 



