(05 > 
VI.—Theorems relating to a Generalisation of the Bessel-Function. By the Rev. F. H. 
Jackson, H.M.S. “Irresistible.” Communicated by Dr W. Prpptiz. 
(MS. received February 17, 1904. Read March 21,1904. Issued separately May 27, 1904.) 
dt 
In this paper, theorems which are extensions of the following, are discussed :— 
Jo(a+y)=I(x)I oy) — 25,(@)I (y) + 2 (a)Jo(y)— - ee eee ad inf. i : ie 5 (a) 
TUR GAs AH GOS See Ce ad inf. , : a ce) 
Jn(2) = ( aa Ue { Jom+n(%) oe i ar zs Ea S eee ae pies 5. yey (x2) — coro en } (y) 
J_,(2) == ( = 1)"J,(x) 
rs 9 n+e d” , 
Tn4i(2) oe ( = 1) ( = aay" { = \ . - 2 -! . . hs (8) 
We define J, (A, x) as 
T=0 
airy CED \n(2)n-er 
[wt+r]iisD([m+r+1]) or IL,([z+7]) 
n is 
Lelie 
(nr oe [lee teeth 
Artery ln+2r] 
In this expression 
The function II,([m]) is defined in the previous paper (Trans. Roy. Soc. Edin., vol. xli. 
parti.). If. be changed to 2A in J,,;, the function will then be more strictly analogous 
wo J,,. 
In Weierstrassian form 
ee tian Uae eas 
TEN wee hee als a 
= I 1 sabe 
P=1+ 2] + El a EO —lo ip 
{2n}! = (2),T,([m+ 1]) 
being in the case of n positive and integral 
{2n}! = [2] [4][6] .. . [27] 
MAO). 2 ste 2 2! 
analogous to 
This notation enables us to write shortly 
— yarn es 
ie ( prt2r] 
1 Dine ar}! ary 
TRANS. ROY. SOC. EDIN., VOL. XLI. PART I. (NO. 6). 17 
