THEOREMS RELATING TO A GENERALISATION OF THE BESSEL-FUNCTION. 111 
and the product 
Tin) ala) 
a { de rte ; rt? 2r J Nae 4 in Lepr e i \ (29) 
{2n}! {2n+2}! (2}1" * On + Br yl {Qr}) Vere 6 (QnF2rjifarh) “os 
From series (28) we are to form a new series, of which the successive terms will be 
homogeneous in AA, and of degrees 0, 2, 4,6,..... Ve i eee respectively. 
The first term of (28) gives rise to the constant, unity. 
The terms of the second degree arising from J o,4); are 
2 Ba rk 
“prep ay 
The term of the second degree arising from J) is 
[4] 
(2) {2}! {2}! 
There are no other terms of the second degree ; the sum of these terms is 
1 [4] 2) 2 
{2}! ray) oe rary, 2 1+ p7Ay } 
1 mie. 
=e 2+ (p? +1), +2°%? } 
A+AP)A+A) 
(2}P {2h 
Terms of the fourth degree arise only from the first, second, and third terms of (28), 
being respectively 
Mé 2d, 2p? dj *p8 
HH!” BH RHE” (ay OF 
ist ABA, AA 2p! 
Bh TAQ} (aH! * TH}! ray} ee en 
pS] ] MA, 2 
(ay {4}! (4)! 
Remembering that 
{4}!=[4][2] and {2}! =[2] 
we write the sum of (30) 
1 
yey ** tte tapas ape yp te | 
Replacing 
5 by (p?+1) and fn by p*+1 
the expression within the large brackets reduces to 
(AHA )A + p7Ag)(A + PPA, )(A + PA) 
The term of the sixth degree in X, A, I have verified as 
ITE BT ET] | OT ATALAA+ DY A+APIA+ PIAL AEE (831) 
