580 Proceedings of the Royal Society of Edinburgh. [Sess. 
, r/(u) 
sin uv du — I 
m du , 
Jq U 
provided either h(u) is an odd function, or f ■ U -du exists , y(u) denoting 
iq U 
the total variation of g(u) measured from infinity. 
Theorem 12, bis. — If throughout the interval (q, oo ) 
f(u) = g(u)h(u ), 
where g(u) has bounded variation in the whole infinite interval and 
approaches the limit zero at infinity , and h(u) is any summable function 
whose Lebesgue integral , proper or improper, can be expanded in a series 
of the form 
f u h(u)du — B 0 u + H (A n cos k n u + B„ sin k n u), 
J n = 1 
where the positive quantities k n have infinity as unique limit as n 
increases , and the series whose general term is | A n | + | B n | converges, then, 
if f" f (V I d u exists, 
* J q U I 
I dv j f(u) sii 
Jo J i 
provided the coefficients A n are identically zero, or I ^ U du exists, y(u) 
J < i u 
denoting the total variation of g(u) measured from infinity. 
This last theorem includes all the Theorems 8-12 as special cases ; 
moreover, we may, in the enunciations of those theorems, change the 
monotone function into a function of bounded variation in the whole 
infinite interval, provided we add the condition that I — — du should exist, 
J q U 
y(u) denoting the total variation of function in question measured from 
infinity. 
Section 4. 
On the Fourier Cosine-integral J* dvj” f(u) cos uv du. 
§ 21. As in Section 3, the discussion reduces itself to that of 
Jo do\ fiff) cos uv du. 
The theorems which occur are in striking parallelism to those of the 
preceding section. 
§ 22. Theorem 13.—// J” j f(u) | du exists, then 
Jo dv\ q f(u) cos uv du = 0. 
By exactly the same argument as in the proof of the parallel theorem 
in Section 3 (§ 12), we arrive at the equation 
/* B r B r f(u) 
I dvLf f(u) cos uv du — Lt du f(u) cos uv dv = I ' ' sin B u du, 
J 0 Q — » J q Q=oo J g J 0 J q U 
