( 507 ) 
If we displace the origin to g and if accordingly we call 2 — a = , 
we find for the orbit of £ the logarithmic spiral y' = c€& round the 
point g. If, however, a is real, then z describes the line from z t to 
az 0 -J- ft, containing also point g — -- . Is on the contrary 
mod «= I, then the orbit of : is a circle round g as centre. 
IV. y — 
yx-\-d’ 
where («—df -f- 4£y v °- 
_ («+d)4V («—_ lo {(« 4- <>)+ 
* — ° 9 (« + d)— ~ 9 Mad-dy) 
_ («— d)— Via— d) 3 + 4|3y 
(«-d)+V / («-d) , + 4^y 
2(3 
2^ 
; all complex; 
2 . + ? -1 
We shall take as general case, that p, y, and d 
then x, p, and q will also be complex. 
/w = I + 
From 
z+p-' z 0 4-p- 1 
ensues that for an infinite value of n the point z takes either the 
value —p~ l or the value — g r ~ 1 . We shall call the points z = p 
, xi.. i.-— ot,h wo shall put — p—'=a'. 
+ »• 
(2) 
— — q— 1 the limiting points and 
<r x =9"- 
Thus our equation (2) becomes 
log - = log — - y, + (p + n ' 
z-9 z «-9 
where we have replaced X by ft + *»• 
Let us choose <j and g” as auxiliary origins and let us 
z—g’ = z’ = •>'#»' , z—g" = *" = 9 "d e " , 
we then find out of (3) 
log 1 + — 0") — log - i — iin + w 
Q Q 0 
where by separating the real part from the imaginary wf 
log^y — log - = ftw , (0'—&") — (0'» & ») = v 
