whilst in the case 
validity. 
( 512 ) 
f* = 0 only the second determination retains its 
/<*) = 
log n 
log log z log log z 0 
log a log a 
put log a = f iv, we then have 
log log z = log log z 0 (p -f iv) n, 
/{*)-- 
f = log z 
log q -\- id = {log q 0 -f 3 
from which ensues 
ivn), 
* {cos vn 4- i sin vn ), 
{log q)* + 0* = {{log $ o y 6y\ ,1 
l°9 Q _ l°9 Qo vn — 0 O sin vn j • • • • (13) 
6 log p 0 sin vn 4- 0 a cos vn ’ ) 
Out of these equations follows by elimination of n the orbit of 
For the case a positive,' so v = 0, the second equation passes into 
log (j _ log Q 0 _ 
q = <*K 
The orbit of z is in this case af logarithmic spiral around the 
origin, which is independent of a. 
If mod a = 1, then p == 0, so that the first equation (13) tells 
us that 
(log 9 y + 6* = (log + 6 >.* = t ’ 
or 
This curve is likewise independent of the argument of a. 
The function y = x~ l , which we have regarded on one hand 
under IV A, p = 0, and which then furnished for the orbit of 2 a 
circle, we can also range under the case treated last. If namely 
we take y = x~ as a special easeof^ = tf* {mod. a = l, arg. a = n), 
we then find for the orbit of 2 quite a different curve. 
To this remarkable property of y = sr~ l we hope to refer more 
explicitly later on. 
