562 
Proceedings of the Royal Society of Edinburgh. [Sess>. 
(Ila.) The function h(u) is a constant ; in this case the condition that 
r fi'ijii') ^ /"°° et(u') 
I — — \clu should exist may be omitted, and the condition that / — 
./ q U | ^ J q U 
exists is only necessary in the case of the sine-integral. 
(Ilk) The function h(u) is of the form cos ku or sin ku ; in this case 
/O) 
the condition that 
i 
u 
du should exist may be omitted, and the 
condition that 
/: 
u 
du should exist is unnecessary in the former case 
for the cosine-integral, and in the latter case for the sine-integral. 
(lie.) The expansion of J u h(u)du contains no sine terms, or no cosine 
terms ; in the former case we may, in considering the cosine-integral, omit 
f ^du should exist, and in the latter case we may 
J Q 
the condition that 
u 
omit this condition in considering the sine-integral. 
(lid.) f(u) is expressible as the product of two factors, one of which, 
g(u), is a monotone decreasing function with zero as limit , while the 
second, h(u), is any periodic function whose square is summable, provided 
further that J ^pdu and J 
|f(u) 
! U 
du both exist. 
These conditions differ, as already remarked, from those hitherto given 
in the fact that the generalised integration has been freely used. It has 
moreover been usual to consider only the cosine-integral. With regard to 
the conditions when we are considering the latter integral, it may be 
noticed that there is a slight advance in the statement of condition (ii.) at 
the origin, as previous writers have omitted to remark that f(u) need not 
have an unique limit. The condition (iii.) at the origin includes as a 
particular case that virtually given by Pringsheim. The condition (iv.) is 
here stated for the first time. 
As regards the conditions at infinity, (Ila, b, and c) are due to 
Pringsheim. The general condition (II.) and the special one (lie?.) appear 
to constitute a material advance on the corresponding condition given by 
Pringsheim. 
In obtaining these extensions all turns on the use of a theorem in the 
theory of Fourier’s series, and of a more general theorem in the integration 
of series, recently stated and proved by myself. The reasoning adopted 
is moreover differently arranged from that of Pringsheim, though the 
general line of argument is the same. 
