442 REPORT—1904. 
and from this it is an immediate consequence that the number of zeroes inside the 
circle is exactly N°’, Again, between the circles r = N? and r=(N +1)? 
de 
f(r) = anit +6) + wen +e). 
The 3N?+5N +1 zeroes which lie between the circles are given by the formula 
AN+ 
r=N? + = NP Pies 
= ee). 
3N?+3N+1 
(4=0,1,...8N?+3N), p being very small. A more precise approximation to r 
would not be difficult, but seems superfluous, 
The analysis is practically the same for more general forms of (mn), and may 
be applied to other functions of the forms 
Sy” > ve 
ip(m)} {p() 39’ 
&e. 
B. The function 
fae, New oe 
Si(np + lyn! 
Concerning this function I have arrived at the following results :— 
i. If p and a are positive, and a region, D, of the plane be defined by the 
inequalities —or< as es 6,< an, then 
A 
Sa p NY = Gay"! +€), 
where e, is a function of A, which tends uniformly to zero when A tends to 
along any path inside D, and \~* is real on the real axis. 
ii. If D’ is the image of D in the imaginary axis 
Dispos ly 
f(a, p, 4) = —P*(—A) pf{log (—A)}*-“(1 +.) 
PT 
1 
in D’, (—d)? and {log (—X)}*-1 being real on the (negative) real axis. 
ill, Ifp and a are positive and a integral the zeroes of f(a, p,X) which lie 
above the real axis tend to the points 
far i EG 
(« ey) log (2hm) + (a—1) log log & 
1 
4 = 
+ log Gs [ (2x sti NM oh 37 («+ 5) |, 
where / is a large positive integer. 
iv. If, on the other hand, a is zero or a negative integer f(a, p, d) has but a 
finite number of zeroes, which are all real and negative, reducing in fact to the 
product of e* by a polynomial. I have no doubt that the restriction introduced in 
li. that a is integral is quite unnecessary, and that the formula holds for all real 
values of a save negative integral values, with a slight modification when a is 
negative. But I have not been able to prove this rigorously. 
The function 
ao 
— rn 
= 
nn! 
