588 
Proceedings of the Royal Society of Edinburgh. [Sess. 
or 
(ii.) f(u) is monotone decreasing with zero as limit at infinity, 
or 
(iii.) f(u) is the product of two functions, one of which, g(u), is monotone 
decreasing with zero as limit at infinity, and the other, h(u), is a sine or a 
cosine or any periodic function whose square is summable in every finite 
interval, provided in the latter case 
u 
du exists. 
Still more general forms may be given to the sufficient conditions (iii.). 
Thus the square of h(u) need not be summable, provided h(u ) is summable, 
and the series whose general terms are | A n | and | B„ | are convergent, where 
A n and B w are constants such that 
\ u h(u)du = A 0 ?i + 1' (A ?l cos h n u + B )t sin k n u), 
J n = 1 
the quantities h n being positive and increasing without limit as n increases. 
In particular, therefore, this is true if the series whose general terms are 
ajn and bjn are absolutely convergent, a n and b n being the Fourier con- 
stants of h(u). 
I have not in the present paper thought it worth while to call special 
attention to the case when f(u ) or g(u) is a function of bounded variation. 
This case belongs to the more general case when f(u) or g(u) is the 
algebraic sum of a number of functions belonging to the types above 
specified, multiplied by +1 or — 1 . 
In conclusion, I here give the statement of a theorem which, though 
now well known, is rarely stated in its most general form, and which is 
continually used throughout the present paper A 
“ If s n (x) is a bounded function of ( n , x), and, except at a possible set of 
zero content of values of x, converges to f(x), and g(x) is any function 
possessing a Lebesgue integral, proper or improper, in the finite or infinite 
interval (a, b), then the sequence of Lebesgue integrals \s n (x)g(x)dx con- 
verges uniformly to ^ h a f{x)g(x)dx.” 
§ 2. Theorem. — If f(u) is any summable function in the interval 
(0 , p), having at the origin the unique limit f ( + 0), then 
Lj'Jo dv Jo e~ kvH f(u ) cos uv du = r/( + 0). 
* The statement and indications of proof of this theorem are given in my recent paper 
on the “ Theory of the Application of Expansions to Definite Integrals,” § 4, presented to 
the London Mathematical Society. 
