296 
MR. E. AV. BARNES OX THE THEORY OF THE 
we shall have 
_ IVihA’) 2 .J: -i,). 
r,-\z) - ^{27r:co,y 
(I). 
0 )^ (lues 
1 .. 
(2) Secondly, when ^ ^ lie wdthin the region bounded by i 
,^axes to — 1 
and — 
In this case /I’l = i 1 , 
i ' m — 0. 
^0 = i h 
We shall have then tlie same relation 
r -1 
(X - A 
Wb’) ~ v/( 27 rVd 
])i'ovided v'e take 
yoo((yj, oj^) 
n 1 
c<l -*■ 
V S' 
0 0 — 
I, t ] \ "l + "3 1 / , \ , (2/? + 1) Wo — Wii 
(/t ~r g) mg 71 ioj I -j~ ^■■>) “k-1*-'K 
WjWo ~ 2a)2^(Uo 
+ 
(‘2'U + 1) Wj — Wo 
2 Wo 
log noj^ =F 
{)l F 1) 77i 
Wi 
the upper or lower sign being taken according as I(wo) is positive or negative. 
( 3 ) The third case, when 
w, 
does not 
lie within the region bounded by axes 
(ojj Wo) does 
• I and — Wo, is easily seen to l)e impossible. 
In all other cases we shall have the relation (1), })rovided 
^ 2 ) — ^ n ~ + i") A ;r b)g?i(w^ + Wo) + . log 
0 0” 
710 ).■) 
w,w., 
-^W^W,2 
(2ji F 1) w, — w., , 
+ -^-■ iog no. 
(^)- 
Suppose now that we liad investigated similai'ly the quotient 
Fg ^ ( 3 ; F W., ) 
r.-' (~d ' ’ 
we should have obtained tlie difference equation 
r2~b^ F wj _ ly (,:■ I wp _o„,v,/_ _ 
where yoj (w^, Wo) has the value D, let us say, given by equation (2), except in two 
cases. 
Wo does 
, (w^ 4- Wo) does not 
and — Wj, in which case 
(1)' When 
lie within the region l)Ounded by the axes to — 1 
(ii F 1) 77 1 
y-22 ("n Wo) — D dr 
Wt 
the upper (.)r lower sign being taken according as I(wj) is positive or negative. 
