37G 
MR. E. W. BARNES ON THE THEORY OF THE 
(l, 3*^3)log' “1“ ''0*^3) “1“ (ij “1“ 
And now by mere re-arrangement we may include a with the term n i"i + each 
time that the latter occurs, and we obtain for log Eg ( 2 ), where z = a + n.oy.^, 
an asymptotic expansion of the form 
(1, 2)'^logz + (1, zf -h 2 
A = 1 
§ 81. We can readily extend the previous proof to the case where 2 lies between 
the axes of — and w,,, so that it is given by 
z = Cl — “b n.^oin. 
By writing the fundamental difference equation in the form 
l\{z - OJ,) = r 2 ( 2 ) 
“1 1 “2; 
we readily see that 
log To {z — n^Q)^ + noWo) 
'1 -.W \ EiCr + mowd"!) 
log l a (-) - i log-- 7 —^- 
'Dl.i = 0 
Hj '/U - 1 
V 
log r3(2 — I — oj^, C0.2) +2 2 log (2 — -g W0&J3) 
■Jill = 1 ?n.2 = 0 
+ 
+ — logEo (, 
L 7 /ti = 1 Pl\ ^ 2 / 
— 2 2,mTTLS^'(z —nijO) J a),j) “h 2 27 /^ 771 , S/( 2 -j-^UoOJo | Wj) . 
I \ I ^ 1 I coo) 
2 — I — Wo) + 2 log -^^ 
TOi = 1 
But by the theorems just quoted in the previous ])aragraph the three expressions 
in the scjuare brackets severally admit of asymptotic expansions in powers of ^ and 
—, whose terms which do not ultimately vanish are typified by 
>C2 
(1, 7if log a -f (1, -a)- 
and whose coefficients all involve a algebraically. 
Thus, by a repetition of the previous argument, log ro(o — ^qw, -fi 7 ioWo) admits 
when z = (I — + n.^cD.^, an asymptotic ex^jansion of the form 
(l.#logZ + (l,2f + 1 g. 
A = 1 
In an exactly similar manner we may show that log r 2 ( 2 ) will admit of an asymp¬ 
totic expansion of tlie same form when l 2 | is laig " Iving between the axes of — 
and Wy. 
