560 Proceedings of the Koyal Society of Edinburgh. [Sess. 
in the neighbourhood of the point infinity, and in the intermediate portions 
of the range of the independent variable separately. It then at once 
appears that the function which in Pringsheim’s work is expressible in the 
form of a trigonometrical series, need not be differentiable, whereas 
Pringsheim requires it to possess very restricted properties as regards 
differentiability. A very slight re-arrangement, therefore, of Pringsheim’s 
reasoning enables us to increase the scope of his results. Again, through- 
out Pringsheim’s paper the reference is to ordinary integration, or to the 
various extensions of this concept known before Lebesgue had so materially 
increased its generality. Here a slight change of wording, and a reference 
to theorems recently discovered, allows us to interpret Pringsheim’s 
theorems, so that they may be applicable to the more generalised concept 
in question. 
The desirability of reconsidering the subject from this point of view is 
responsible in part for the length of the present paper, whose original 
object, however, was that of extending in an important particular 
Pringsheim’s result, as to the condition to be satisfied by the function in 
the neighbourhood of infinity, and so, en passant, to confirm its validity. 
Occasion is also taken to expose corresponding results for the repeated 
integral 
^dv^/(u) sin uv du. 
The interest of this integral lies in the fact that it is the analogue of the 
very important but little-studied trigonometrical series obtained from a 
Fourier series by interchanging the coefficient of cos nx and that of sin nx, 
and changing the sign of one of them, for all values of n. 
I now give for convenience of reference a table of the results proved in 
this paper, at the same time indicating which of them are to be regarded as 
new, and to what extent. 
Sine Integral. 
Result. 
j dvj f(u) sin uv du = j jW u ^ du. 
Condition in any Finite Interval (p, q). 
/; 
f(u) - f( - ll) 
2u 
du exists as a Lebesgue integral. 
Cosine Integral. 
Jo dv f f(u) 
cos uv du = ^ (/( + 0) + /( - 0)). 
A 
Condition in any Finite Interval (p, q), 
NOT CONTAINING THE ORIGIN. 
J^f(u)du exists as a Lebesgue integral. 
Alternative Conditions at the Origin. 
(i.) f(u) has bounded variation in some 
interval containing the origin as internal 
point : or 
