578 Proceedings of the Royal Society of Edinburgh. [Sess. 
k n , having, as n increases, the unique limit infinity ; the argument will 
remain valid, with the sole change that the choice of the integer m, im- 
mediately after the equation (1), must now be such that, for all values 
of n A m, 
K - B > \k n . 
We thus have the following extended form of the theorem : — 
Theorem 11, bis.— If for all values of u in the open interval (q, oo ), 
f(u) = g(u)h(u), 
where g(u) is monotone decreasing with zero as limit at infinity , and 
h(u) is a summable function whose Lebesgue integral, proper or improper, 
can be expanded in a series of the form 
J M h(u)du — B 0 w + % (A n cos k n u + B ;l sin k n u), 
n= 1 
where the positive quantities k n have infinity as unique limit as n in- 
creases, and the series 2 ( | A n | + | B n | ) converges, then, if / 
n=l J q 
r 
f(u) 
/q 
u 
du exists, 
rm 
J Q U 
du , 
zero. 
I dv I f(u) sin uv du 
Jo J q 
/*" g(u) 
provided I ■ v du exists, or the coefficients A u are identically 
Jo. U 
°0 
§ 19. If we make the more stringent condition that the series 2 1 a n | and 
n= 1 
OO 
2 1 b n | both converge, where 
n= 1 
d’nji'n Bn 5 <Uk1 b n fk n A n , 
the series whose general term is 
a n cos k n u + b n sin k n u 
will converge uniformly, and may therefore be integrated term-by- 
term ; thus we may take h(u) to be the sum of this series, and it will 
then be a bounded function of u, so that 
/: 
g(V 
fi 
/(u) 
J* 
u 
du will exist, provided 
u 
du exists. 
Thus we get the following corollary : — 
Cor. — If the series whose general terms are respectively | a n | and | b n | both 
converge, and h(u) denote the sum of the series whose general term is 
a n cos k n u + b n sin k n u, 
k 1? k 2 . . . . k n being any succession of positive quantities increasing 
without limit as n increases, then 
