MATHEMATICS: HARDY AND LI TTLEWOOD 
87 
for every positive €. The publication of WeyFs work had led us to a 
re-examination of this question and to the following theorems. 
A. // (i) X„+[„6] - X„-^ CO , 
(ii) ki|2 + k2h + . . . + knh = O(;^^+0, 
for every positive e, then 
(iii) sAa)=^aueM = 0{n^+') 
1 
for almost all as and every positive e. 
B. 7/ (i) is replaced hy (iO Xn+[f»/8+€] - X„-^ 0 < /3 < 1, /to 
(iii) wa^' 5e replaced hy 
(iii') 5„(a)=0(«-^ + *). 
To these two theorems WeyFs forms a completing third. It should be 
observed that (i) is certainly satisfied if Xn+i — X„ > c > 0, and in 
particular if X„ is always integral, and (ii) if an = 0{n^), and in partic- 
ular if =1. 
If X„ is an integer, and we separate the real and imaginary parts in 
the equation (iii), we obtain a theorem concerning a particular system 
of normal orthogonal functions for the interval (0, 1), viz., the functions 
V 2 cos 27rXnX, V 2 sin 27rX„:r. Our argument is then directly exten- 
sible to a general orthogonal system, and we are led to a new and inter- 
esting proof of Hobson^s^ theorem that if <j)n{x) is any normal orthogonal 
system, and 2 | ^ convergent for some positive 8, then XCn<l>n (x) is 
convergent almost everywhere. 
WeyFs h3^othesis concerning X^ asserts, roughly, that the increase 
of X„ is appreciably more rapid than that of (log n^. It is easy to see 
that this hypothesis cannot be capable of much wider generalisation. 
For, when Xn = log w, 5„ is definitely of order n. It seems probable, too, 
that the index J (1 -|- |8) of Theorem B is the correct one. 
5. We conclude by correcting an error in our recent note. The re- 
sults concerning the special case p = 0 are stated wrongly. It is not 
true that, when p = 0, /(s) and Sn are bounded; all that we can assert 
is that they are of the forms 0 (log ^ and 0 (log n) respectively. 
That / (2) should be bounded would contradict a general theorem of 
Fatou,' in virtue of which a bounded function must tend to a limit, for 
almost all values of 6, when z = re^ tends to the circle of convergence 
along a radius vector. The error has no bearing on the general case. 
