344 
MR. E. W. BARNES OX THE THEORY OF THE 
And now, since x 1 vanishes for Jc 2 values of s, we see that its expansion 
A\ hen : is large must be 
-1 /r+3 (-S) I Pi+3 (^) I 
.-.i+^+2 V" ,>s+/f +3 ~r • • ■ • 
The series for (s, a|w^, (o.i) is therefore convergent with 
00 00 
7^1 = 0 7 / 1.2 = 0 
_ 1 _ 
(a + 
It is the]i convergent when .s + A: + 2 ; > 2 or il ( 5 ) > — k, js] being tiiiite. 
We have then obtained a solution of the difference ecpiation 
J [a + -f Wo) — J {(t + Wj) —J{ci + Wo) + / (a) — a \ 
where a has its principal value with respect to the axis of — ("1 + Wo), which is 
valid for all values of s, a, w^ and Wo. 
§ 58. The identity of the function ( 5 , a | w^, w,) just defined with that previously 
employed is easily seen. 
From the mode of formation of oS_,._/: (a ] W[, Wo) it is evident that when a is positive 
with respect to the w’s, we have 
('^ 1 Wo) = the sum of the first (k + 2) terms in the expansion in powers of - of 
^ g2M7riS 
i-i 
and therefore 
L (1 — 
dz , 
2 ^— 00 (^1^1) ^ 2 ) — 
/r(i - s) 
c-«-- ( ~y- 
jTT 
'L (1 — c-“r-)(l — 
dz. 
Therefore when is large we have the asymptotic expansion 
of 
{« + + t) W^ I Wp Woj - 
1 
- 1 . 
CO, 
1 1 2u “j- W| T Wo 
{s — 1) (s — 2) WjWo (p^Wj)^”" s — 1 2wpt)o ( 2 }nco^y~^ 
+ 
{pncod‘"'''^~^ 
1 _ V /_yo-i s (s + 1) . . . {s + Hi — 2) + wd 
(pnco^y-'^ „Zi 
III 
a formula which may be jiroved to be true for all values of a by a term-by-term 
ex})ansion of the series for pS_o_oo{rt + + l)wjw|, wo]. 
Now from the expression for {o{6‘, «|wj, wo) given in § 57, we see on takiipg the 
(pn + f) fii'st values of and the {qii + t) first values of ??!o, that 
{o (.S', a ] W|Wo) = 
]ni qii 
— oS_iy,[rt -f- + l)w^ + 
= 00 _oi, = 0 /»,, = 0 ('f ■E /^pWj -f- 7/OWo)^ 
+ pS_o.y, [a -f- ( 'pii “h I ) W| j -)- oS_jy,(« + {([It -b 1) Wo} 
