324 
applying to the supposition that z, moving along a radius of the 
circle |z| = 1, approaches a given rational point on the circumference. 
Moreover the investigation will serve to add some results to those 
of Knope concerning the function A (2). 
Qn1p 
1. Supposing p and q to be integers prime to each other,e 7 = 67 
is a rational point of order q on the circle 2 = 1, and taking 
O<r<1 we have 
n= any h=q—1 n= angth hp 
N= 2 bag ee eh es 
( ) n=1 Targ h=1 n==0 Mer? 1—arat+h Ohp 
Now obviously we have 
qany ung h=q—1 yng Olp 
PT 1—arnd zi lars Olp 
and 
a oe (php 1 
a iO ar oa 
hence we get 
n= OD wey 
h=q—1 
N(eO) 4 B bm Faes = Uiep,  W 
ange 
where the coefficient 6, is arbitrarily chosen and where 
nen aergth Ohp ang Alp 
Oe a Pugth 7 erat Gp" 1 ana Ol (2) 
The right ae can be transformed, we may write also 
U;, (a, p) = — (joes Bagh - res we a 
jn (l—anath Olp) (1-- ang Of) (3) 
naz 00 ang Alp 
by »—b REET ER 
a = Cae 0) — xq Olp 
In particular, taking bj =1 also 6, =1 we find for the function 
of LAMBERT 
h=q-—1 
L (7!) an q L(x) == 5 (ga!) = = Th (,p), © Bt. TE (4) 
ill 
where 
= "5 | angth Ohp ud Arp 
h (7, DS <9 ras | ies angth Alp 1—arg Olp Rie 
5 
n= ang Olp ( ) 
= (le) 
et ee 
En (l—anath ol) (1—ar9 Ohp) 
Now we may observe that if O <w <1, the moduli of the factors 
