But 



dz 



of small Fractional Order. 



. ZV . J v (z) = Z?-\zJ p ' + P J P )=f . Jp-^Z). 



20.'5 



Hence, the roots of (5) are simply related to those of 



J*_i(V). Since s=Pj £° r values of m from —1 to co , 



^ v y ?n -4- - 



p — l = n ranges from to —1. To solve equation (5) 



a short table of the roots of J n (Y) = is required, n having 



the limits and — 1. A table of this kind, however, like 



that of J»'(s) is inconvenient for calculating other roots by 



linear interpolation, especially when the order n is near 



the latter limit. This difficulty does not present itself if 



the values of p^/^n + l) are tabulated. In constructing the 



following table, the first roots of J n (z) were calculated in 



nine cases, when the order n increases from —1 to by 



intervals of ^, and from these the values of p 1 2 /4=(n-{-l) were 



found. The table was completed by interpolation to fifths: — 



n. 



Pl a /4(»+l). 



n. 



Pi 2 /4(m + 1). 



n. 



pr/4(»+i). 



n. 



Pi74(»+1). 



-1 . 



.. 1-0000 



-1 • 



.. 1-1204: 



i 



.. 1-2337 



-x • 



.. 1-3417: 





1-0124: 





1-1320: 





1-2447 





1-3523 





1-0248 





1-1436 





1-2557 





1-3628: 





1-0370: 





1-1550: 





1-2666 





1-3733: 





1-0492 





1-1664: 





1-2775 





1-3838 



7 

 — 8 * 



.. 10613 



-1 



.. 1-1778 



_3 



8 



.. 1-2883 



i 



~8 • 



.. 1-3942 





10733 





1-1891 





1-2990: 





1-4046 





1-0852 





1-2003 : 





1-3098 





1-4149: 





1-0970 





1-2115 





1-3205 





1-4252 : 





1-1087 : 





1-2226 : 





1-3311 : 





1-4355 : 



-I- 



.. 1-1204: 



i_ 



.. 1-2337 



1 

 4 



.. 1-3417: 



. 



.. 1-4458 



Sir George Grreenhill * has recently called attention to the 

 practical importance of the Bessel-Cliffbrd function, M (#). 

 Since this function can be expressed in terms of the 



n 



J functions, x* . G«(#) = J n (2 */d), the roots of Q n (x) are 

 easily calculated from thos« of J»(j8r)=0 and the roots of 



— C»(«) from those of J, l+ i(V)=0. This readily follows 

 dx* 



from the above relation. 



# "The Bessel-Clifford Function and its applications." Sir Georgi 

 Greenhill. Phil, Mag. vol. xxxviii. Nov, 1919, pp. 501-528. 



