7o 



Tables 36 and 37 

 CYLINDRICAL HARMONICS OF THE OTH AND 1ST ORDERS 



TABLE 36.— 4-place Values f or x = 4 TABLE 37. 



(a) 1st io roots (Rm) of J (x) — o; Ji(R m ) 



Higher roots may be calculated to better 

 than I part in 10,000 by the approximate 

 formula Rm = Rm-i + w 



Rt = 2.404826 

 R 2 = 5.520078 

 #3 = 8.653728 

 R< = 11.791534 

 Rs = 14.930918 



Re = 18.O/I064 



R 7 =21.211637 

 R s =24.352472 

 R 9 = 27.493479 

 Rv, — 30.634606 



A (/?!>= 4- 0.5191 

 A (fo ) = — 0.3403 

 A (/?»)= +0.2715 

 /, (Rt)= — 0.2325 



A (/? B )=+ 0.2065 



A (R,,)= -0.1877 



A (/?t)= + 0.1733 

 A (R»)z= — 0.1617 

 Ji (R s )= + 0.1522 



A (Rm)= — 0.1442 



dJo(x) 

 (b) 1st 15 roots of A(.r) = — -p- -=o 



with corresponding values of maximum or 

 minimum values of Jo(x). 



Higher roots may be obtained as under (a). 

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

 tion of Bessel's equation, 



The general formula for J n (x) is 

 00 



(-l)'X T 



2 n+2s TrsTr(n + s)' 



when n is an integer and 



Jn+l(x) = Jn(x) - Jn-l(x), 



dJ (x) 



and 



A(.v) 



dx 



/_„(*) = (-i)«A(s). 

 Tables 36 to 2>7 are based upon Gray and 

 Matthews' reprints from Dr. Meissel's 

 tables. See also Reports of British Associa- 

 tion, 1907-1916. 



Smithsonian Tables 



