290 
MR. E. W. BARXES OX THE THEORY OF THE 
and where h = d: 1- the np})er or lower sign being taken according as 1(a)) is positive 
or nen’ative, Avhen 2 does lie within this region. 
(.)n differentiating, we have the derived expansion"^^ 
+ « 1 w) = ^ (log 2 + ' 2 l-m) 
CO 
+ terms wliich vanish when !2| liecomes intinite. 
the ])rincipal value of the logarithm l)eing again taken, and h being determined as 
before. 
Substituting in the expression for 'y.-i(a)^, o)^), we have 
(a)|, 0J.2) d- 2 - = Lit 
= X 
CO^ 
'll 1 1 
■ tog hoj.i “h — ncoy 
CO .T ' 0) 
— log n . (a)| “h ca.i) 
_ X X' ffl 
= 0 m., = 0 11 ’ 
'IJCTTI 
&).i 
where Jc = 0, unless 
a)| does ^ (o)] -j~ w.-,j cioes 1 
” (a)| + oi.d does not j aij does not J 
("] + oio) does] 
lie in the region hounded by lines from the origin to —1 and — o).,. [It is understood, 
of course, that tlie principal values of the logarithms are to he taken.] 
When, as in the tigures 1 lie within the region of exception, k 
^ (oji + o),.) does not! ^ 
= ih i, the upper or lower sign being taken according as I(a)^) is positive or nega¬ 
tive. 
From the diagrams, we see at once that it is impossible that should | 
a)^ should not j 
lie within the region hounded l)y the lines from the origin to —1 and o;^. 
Fake now m = l\ tliat is to say, let m he such that we have m = 0, unless 
coj does j region of exception, and in = dz 1 according as 1(a);) 
(•^i + a).i) does not! 
is positive or negative, when this exceptional circumstance takes place. 
* According to i\L Roincarcf we may not in general differentiate ;in asymptotic expression. The one 
in cjnestion, however, may be readily established Ip' the methods emplo^’ed for log ri(: - 1 - a \ w). 
