330 The Hon. J. W. Strutt on Bessers Functions, 



where m is an integer; but the smaller roots deviate greatly 

 from these values. 



If +p denote the roots of J„(2') = (exclusive of zero), 



whence, from (2), 



^/ z^ z' 



° L 2{2n + 2) "^2.4. (271 + 2) (2?i + 4) 



z_^ 1 



2.4.6. i2n + 2)(27i + 4)(2n + 6) + * ' * J 



=iog{i-i:}+iog{i-|}+... 



Expanding both sides in rising powers of ^s*, we find, on equating 

 coefficients, 



sl = _J 



V 1 _ 1 . 



^p"" 16(71 + 1)2(^ + 2)' 



For example, if 7i = 0, 



^;>2~4' V~32^ 

 and, if 7i=l, 



^ I __ 1 ^ i _ 1 



2-8' ^^4-3g4 



In a similar manner, if ±q denote the roots of J'„{2') = (n not 

 being zero). 



If 71 = 0, 



q^ 4;z(w + l) 



It should be noticed that a well-known theorem is included 

 as a particular case. If 7i = |-, Ji a sin 2", and 2 — = -. Since 

 we know that the roots are of the form +m7r, we infer that 



7r2 ^ m2 "■ 6* 



Very simple relations exist between the functions of neigh- 

 bouring orders. As I shall have occasion to refer to them, I 



