318 
MR. E. W. BARNES ON THE THEORY OF THE 
Th Is series is siimmable by an almost evident modification of the application of 
Boeel’s ideas, which was employed in the “ Theory of the Gamma Function,” § 38. 
It is thus the asymptotic equivalent of the sum 
q,i 1 
V _ t _. 
iiii=o i)i2=o ’ 
and it satisfies Poincare’s"^ criterion for asymptotic equality that the difference 
between this sum and the first m terms of the series has its absolute value less than 
P 1 1 
a quantity of order ■ 
§ 39. The function a\oi^, co.^) has been dehned, and the asymptotic equality (A) 
has been deduced only for the case in which the real parts of a, oj^, and co., are all 
positive. 
It is natural to try and use the equality to define the function for all values of 
u, and CO.,. 
In the first place, it is evident that when ff(coj) and ure both positive, the 
equality (A) holds for all values of o, for the various terms of the sum and the equiva¬ 
lent asymptotic series are continuous for all hut an enumerable number of values of 
o, co^, and CO,,. Hence in this case ujcoj, co,,) may he defined as the term inde¬ 
pendent of u. in the equality. 
So also when .§ is a real jiositive or negative integer, the equality will hold for all 
values of a, coj, and co.,. 
But when .s is not an integer, tlie various terms involving s in their index are 
multiform functions, and to ensure uniformity we have to assign definite cross-cuts 
to the logarithms which arise in the ecjuivalent exponentials. When, as under the 
limitations for which the ecjuality (A) has been established, these ci'oss-cuts are 
formed by a line outside the smaller angle between the axes of coj and co,,, the expan¬ 
sion is perfectly valid ; but when the common ci'oss-cut lies within this angle, terms 
arise similar to those which occurred in Pai t II. of this paper, which are multiples of 
2771, and involve rt. 
And, therefore, if we attempted in this case (see the second figure) to define 
a I coj, coo^ as the absolute tei'in in an asymptotic ecpiality such as (A) ^ 38, where 
for conqilex values of s the principal value of each term is taken, we should ultimately 
* PoiNCAKE, ‘Acta Mathenialica,’ vol. 8, pp. 295-044 3 ‘ Mecaiiicpie_CeE^ste,’vol. 2, pp. 12-14. 
