of small Fractional* Order. 203 



The roots p 1 of J n (a?) for values of n between —\ and —1 

 are tabulated in the last sectiou. From the five place table, 

 the following values of p ± have been found : — 



JO), 1*8663: J (a), 2*1423 



-i -* 



J{(x% 2-9026 J^v), 2-6575 



Roots ofJ n '(x). 



Although of less importance in the solution of problems 

 in physics and applied mathematics than the roots of 

 Jn(ie) = 0, the values of the argument making J„(a?) a 

 maximum or a minimum are required in some cases. When 



the order n is negative, the function -~ . J»(V), where 



° ax N J 



— l<n<0, has a pair of imaginary roots. From the 

 identity 



n + 2 x 2 , ?z + 4 «r 4 



n '4(71 + 1) n "4. 8(n + l)(n + 2) 



-(i-S^^X 1 -^ (3 > 



where a^, ct 2 , cr 3 ... are the roots of J,/( t y)=0, we find that 



_ 6 _ n 3 + 16n 2 + 38n + 24 



Z ' °" s ~~ 4 . 8 . n 8 (n + I) 3 (n -h 2)(n + 3) ' ' ' U 



As in the case of J»(j?), only the approximate values of 

 <7 2 = 7r(2?i-t-9)/4, 0- 3 ==7r(2rc + 13)/4... are required to give 

 <r x with considerable accuracy. A table of the roots of 

 Jn't^)? especially the imaginary roots, cannot be conveniently 

 used over the full range from n = — \ to +^ for determining 

 intermediate values by interpolation as the second and third 

 differences are considerable. This difficulty can be removed 

 by tabulating 8 = <r 1 /^/2w, where linear interpolation only is 

 needed. The root a x is then easily calculated from the 

 value 3 sj^ln. 



For small positive or negative values of n, between — £ 

 and -\-\, (4) gives the following expression for a x : 



/ 1 , 9 23 2 , 35 3 \1 ,« 



For the second root of J n f (x)» MacMahon's formula gives 



