586 
Proceedings of the Royal Society of Edinburgh. [Sess. 
Cor. l| — 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, 
the positive quantities k n increasing without limit with n, then if 
f ^ | du exists, 
J q U 
J 0 dv^ q g(u)h(u) cos uv = 0, 
g(u) denoting a function of bounded variation in the whole infinite 
interval, having zero as unique limit at infinity, provided, in addition, 
either the coefficients b n are identically zero, or j ^^du exists, y(u) 
being the total variation of g(u) measured from infinity. 
If we write (u + x) for u, c n cos r n for a n , and c n sin r n for b n , we get, as 
a second corollary, Pringsheims result * : — 
oo 
Cor. 2. — If the series^ c„ converges absolutely and h(u + x) denote the 
n=l 
sum of the series whose general term is 
c n cos { k n (u + x) + r n ), 
then if g(u) is a function of bounded variation in the whole infinite 
du exists. 
r 
g(u) 
J q 
u 
J 0 dv\ q g(u)h (' u + x) cos uv du — 0, 
provided, in addition , either r n = 0, (mod. tt), or f ^^du exists, y(u) being 
J q U 
the total variation of g(u), measured from infinity . f 
* Loc. cit., pp. 399-403. Pringsheim’s proof, as printed, appears to require further 
justification. At the top of p. 402 the author divides a certain infinite summation into 
two parts : (a) the sum of the first n terms, and (b) the remainder. He then draws the 
conclusion that the part (a), being the sum of a finite number of terms, behaves in the 
desired manner, when the quantity B, which also appears in the summation, increases 
indefinitely ; n has, however, previously been chosen so as to satisfy two inequalities, viz. (25) 
and (26) at the bottom of p. 400, of which the former inequality makes n increase in- 
definitely with B. Again, the conclusion, drawn later, as to the remainder also depends 
implicitly on the inequality (25), since it involves explicitly (27) and (28) ; thus the 
condition (25) cannot be simply dispensed with at this stage of the proof. 
+ The condition that f exists is stated by Pringsheim (p. 397, referred to on 
p. 400 as Condition b ) to be sufficient ; see above, footnote to § 20. The alternative condition, 
r n = 0, is not mentioned by Pringsheim. 
Added 2nd May 1911. 
Professor Pringsheim asks me to say that, in consequence of my criticisms, he is amending 
his proof, and that the corrections needed will shortly appear in the form of a Note in the 
Mathematische Annalen. 
(Issued separately, July 11, 1911.) 
