26 THE REY. F. H. JACKSON ON 
which is 
: Aer 2r glint? 2r) or = oe i lg eae: =| wn == ae pe mes ili 
] ; — 2. = s eo seen d6 
alae er ra) thy ey) 25 ee) 
Ro eo a = Sas 
which is identically zero. 
The only term which does not vanish is the first term in J,,(xA) so that we have 
identically for all values of 7 
Nal) 
[7]! (2)n 
If in 84 we had taken J,,,(#A) as 
— J in(wr) oe Viner @” ) + Biel (cl esc aiek ais. . . (87) 
(2), xu 
n+?r, ltt 27] 
Pa SCN Ne 
the signs in the terms of (86) would not have been alternately + and —; (86) would not 
then be zero but + 
Df 92 1) (p?- 24] ) Netrglrt2r] 
ELS Nee 
Another expansion « in terms of the P functions is 
[2m]! A“ad”) P [2x — 3] [2n—1] [2n-7] 
a | hin EP Pin—9 + p® ie ae ; 88 
[n]! [m]! (2), [n] [2] [n—2] [2] [4] [n—4] ( ) 
analogous to 
2 2n -3 2n-1-2n-—7 
mini 2” Bae poy Pho + at sgt es 
9. 
Various interesting theorems have been obtained with respect to Bessel’s Functions 
_— 1 
when the variable is not « but ./x. The analogous theorems for J,,,(a?+1 *\*) are given 
in the following work. It is well known that 
f{@ ail —_ 
ae 2J( va} = —4)"a" a re ( /x) L : (89) 
(ihe —- 
2 Latif} = QF In B80) 
i) d d d 
liet’ D™ denote ee _ a8 
t ( 1 dae") dare dav") d(?") ( ) 
Then 
WW 1 Nace mitt pi 
pi {x Ne ai “Py. mx 2A#) ; = pr ce "Tinamn( ein!) . f (92) 
which reduces to (89) when p=1 
et a 1 
Ne Wier) = 
0 [7]! [n a5 r|! (2) (2) 40 
