5g Tables 86-86. 



CYLINDRICAL HARMONICS OF Oth AND 1ST ORDERS- 



TABLE 35. — 4-place Values for x = 4.0 

 to 16.0. 



TABLE 36. — Roots. 



(a) ist 10 roots of Jo{x) = o 



Higher roots may be calculated to better 

 than I part in 10,000 by the approximate 

 formula 



Rm = -^m-l + T 

 i?l = 2.404826 

 R2 = 5.520078 



Ri = 8.653728 

 Ri = 11.791534 

 Ri = 14.930918 

 jRe = 18.071064 

 i?7 = 21.211637 

 Ra = 24.352472 

 R9 = 27.493479 

 Rio = 30.634606 



(b) ist IS roots of Ji{x) = ■ j = o 

 ax 



with corresponding values of maximum or 



or minimum values of Jc{x). 



Higher roots may be obtained as under (a). 

 Notes, y = Jn(x) is a particular solu- 

 tion of Bessel's equation, 



The general formula for Jn{x) is 





(-i)«a;« 



or 



2'»+2»7r5 7r(«+i)' 



, (-i)«a--"+^' 



when n is an integer and 



2» 



and 



/«+l(^) = —Mx) - /fv-l(«), 



Mx) 



dJojx) 

 dx ' 



Smithsonian Tables. 



7_„(«) = i-i^Mx). 

 Tables 35 to 36 are based upon Gray and 

 Matthews' reprints from Dr. Meissel's 

 tables. See also Reports of British Associa- 

 tion, 1907-1916. 



