( 24 ) 
Mathematics. — “The continuation of a one-valued function, repre- 
sented by a double series.” By Prof. J. C. Kiuyver. 
In his paper ,Ueber die Entwickelungscoefficienten der Jemnisca- 
tischen Functionen” (Math. Ann., Bd. 51, p. 181) Mr. Hurwitz 
called attention to the perfect analogy between the Bernoullian 
numbers B, and another class of rational numbers Z, occurring as 
coefficients in the expansion of the particular elliptic function pu, 
whose fundamental parallelogram of periods is a square. 
It is possible to carry still somewhat further this analogy. In 
fact, the numbers B, are closely connected with the values of the 
integral transcendental function (1 + e-7) f(z), which correspond 
to positive integer values of z, and we will show that the numbers 
of Hurwrrz admit of a similar interpretation. 
If we consider the doubly infinite series 
1 
fo (med Hm!) * 
—m 
a | Pe MED IER 
+ m 
the ratio @'/o being a complex quantity the imaginary part of 
which we assume to be positive, it is known that this series con- 
verges absolutely as soon as the integer n>2%. Changing the 
integer » into an arbitrary real number a>2 the series is still 
convergent, a determinate value however cannot be attached to its 
sum, until the amplitude of each separate term is fixed in some 
way or other. In order to define this amplitude without ambiguity 
we draw across the plane, containing the network with the vertices 
mo mo, a straight line or barrier leading from the point 0 to 
o and passing through the points @, 20, 30,... Then, having 
fixed once for all the amplitude @ of the stroke w, we agree to under- 
stand by the amplitude of mo + m'o' the angle 0, augmented by 
the angle through which the barrier is to be turned in a positive 
direction till it coincides with the stroke mo m'o'. According to 
this agreement to every real value of a> 2 belongs a determinate 
value of the sum of the series, moreover it is easily inferred that 
its convergence and its one-valuedness are not impaired when the 
real exponent a is replaced by a complex quantity z= # + 7 y, pro- 
vided we have # > 2. 
