342 SECTIONAL TRANSACTIONS.—A”. 
Zo = 7918, . . . . , Whereo<r, S%S .. ., be the zeros of f(z), and let n (r) 
denote the number of zeros for which |z|<r. Then if f(z) is of order p we may 
write 
z 
z 
US BLA pean ties 
f(z) = ett Shel BEI Spe”. (re) on - Pr? 
Zz, 
] 
where p = [o]. 
Tf p is not an integer, the type of f(z) depends only on n(r). In 1905, Lindelcf 
showed that if p is an integer, the type of f(z) depends not only on n(r) but also on 
1 1 
S(r) = ep + Fy > 2 
PEt 
In particular, if n(r) = O(7?) and S(r) = O(1), but one, at least, of these relations is 
not satisfied with o instead of O, then f(z) is of mean type. 
With the improved methods now available we can obtain more precise relations 
between n(r), S(r) and | f(z) |. For instance, if n(r) = o(7?) we have 
log f(z) ~ S(r)z 
in a set of density 1. This is proved by splitting up the canonical product in an 
appropriate manner. If, on the other hand, n(r) is large compared with 
lim log | flrei®) | , 
Se 
then S(7) tends to a limit. This is proved by the method which Titchmarsh used for 
a special function. The method involves a formula for mean values of the series 
Xz,’ and a Tauberian argument. The formula is very similar to Jensen’s formula 
for n(r). 
Dr. E. H. Linroor.—Fourier Series of Almost-periodic Functions. 
A short account of the present state of the theory. 
Mr. R. E. A. C. Patey.—A Remarkable Series of Orthogonal Functions. 
We define Rademacher’s functions ¢,(¢) in the following way : 
glt)=+1, 0<t<$; golf) = —1, 4<t<l 
@olt + 1) = golt) > 9,(t) = po(2"t), n=1,2,.... 
From Rademaclix’s functions we form a set of functions ),(é) : Yo(t) = 1, and 
if n= 2" 4 Oat, +2", n=1,2,....5 
then Vrlt) = Prilt) Prolt) » - ~~ Pna(t)- 
These )-functions have been defined in another way by Walsh, and their properties 
have been discussed by him, and later by Kaczmarz. They compose a complete 
normalised set of orthogonal functions. 
A function f(t) of class L may be formally expanded in the interval (0, 1) as a 
series j 
f(t) bt Sahn 
0 
mies me | F(OY, (td. 
There is a close analogy between the behaviour of series of this form and that of 
ordinary Fourier series, as there is between the behaviour of series of Rademacher’s 
