[HARKNESS] THE ALGEBRAIC BASIS OF CERTAIN BESSEL SERIES 117 
§ 17. The ie (37) gives as special cases 

if 
Int = ual ae ee +255 . He) Je] 
ome 
Lt 2 or (ir, J 
= 2 => ei 
IR Ue = nee iy DD; Be As2 Jn In J 
When we add by columns we find that 
2 5 
Jar os Jn+3 Ve a nie ne 
Se a SEE oo Us 2 
DEN end TA Le 1; 
37S ins TZ 371$ TT 
SEA we have arrived at an apparently new formula for J°,, 
namely 6 
Jon =2[Jatr Jus — Inga Jn-2 + Int Jus — Jute Jn-a...] (48) 
2 2 2 2 2 2 2 2 2 
The special case which arises when n=O is deserving of note. 
a the first place 
=2(J at JT ; TJS on A JE sg. a) 
This can ‘be Givpitied By using the formulae (G. and M. p. 28) 
1=J%4+2 J44+2 J2+. 
Jo (2x) => J*5 — 2 JP + 2 es 
It then becomes 
Jo (2%) = 2 [Ja Ja scl Spi) ap cte | (44) 
As regards ant appearance Ge beaks a considerable resemblance 
to the formula (24) for the integral sine function. 
When (43) is compared with the theorem quoted at the beginning 
of § 8, we see that 
2 2 2 2 
and (45) 
2 [Jn 41 Je_,+Iny, Jn_ gto] 
5 
have the same sum 

$ 18. We shall next establish a companion formula to (19) in § 8 
for J2, (2x), namely 
Jo, (2x) = AT, 17,14 7, 3 J,r3 +...) (46) 
On the right-hand side the general term is 
4(—1): 1 [ _. ari es ae 
L Ta+s) T v+s+2) y+s+2.v+s+3 
n yts—2.v+ts—l.yt+ts.vt+s+1 i Qv+2s “ Qv+2s 
yts+2.vts+3.yv+s+4.7+5+5 AC S JE) 
