of small Fractional Order. 



201 

 for a 



reduces to J_i(.r) = when m~0 and n = l b i.e., 

 homogeneous rod of uniform section. 



The roots p s of J»(a') where the order n of the function is 

 fractional, viz. +j-, + §, + £, and + j, have been calculated 

 by Dinnik * to two places of decimals from tables computed 

 to four places for # = 00 to 8*0 by intervals of 0*2. 



n. 



Pi- 



p 2 . 



n. 



Pi- 



Pi- 



I. 



3 



4 •• 



. 3-49 



6-65 



-1 - 



. 1-06 



4-29 



7-44 



. 2 



3 "■■ 



. 3-38 



6-53 



~~ f •• 



. 1-25 



443 



7-58 



1 



3 " 



. 2-90 



6-03 



i 



— 3 •• 



. 1-88 



4-99 



8-12 



I - 



. 2-78 



5-91 



_JL 



4 •• 



. 2-01 



5i2 



8-25 



Tables of J+i(#) are given by G. N. Watson f to five 



places of decimals for values of x from 0*00 to 2*00 by 

 intervals of 05 and from 2'0 to 8*0 by intervals of 0'2. 



Roots ofJ n (x). 



The first root of J„(a?) is the most important in its applica- 

 tion to physical problems. When n is small, the value of 

 this root can be found by a method introduced by Girard % 

 in 1629 and employed by Lord Rayleigh § in calculating the 

 roots p x of Jo(#) and Ji(a?). 



The sum of the powers of the reciprocals of p { , p 2 , ps . . ., 

 the roots of J»(#) are- expressed in terms of the coefficients 

 of the expansion from the identity 



+ 



2(2^4-2) ' 2.4(2rc+2)(2n + 4) 



(1) 



by taking logarithms of both side?, expanding and equating 

 like powers of x. Thus 



2o- 10 - ln±W (2) 



Ps 2 9 (n + l) 5 (n-+2)\n+.3)(n + 4)(n + 5) V } 



* "Tafeln der Besselschen funktionen." A. Dinnik. Archiv der 

 Math, mid Physik, Band 18, p. 338 (1911) ; Band 21, p. 326 (1913). 



f "The Zeros of Bessel Functions." G. N. Watson. Proc, Royal 

 Soc. A. vol. xciv. pp. 190-206 (1918). 



X Calcul des Derivations, Arbogast, 1800, pp. 56-57. 



§ "The Numerical Calculations of the Roots oi' Fluctuating Func- 

 tions." Lord Rayleigh. Proc. Loud. Math. Soc. vol. v. pp. 119 124 

 (1874) j or Collected Works, vol. i. pp. L90-195. 



