( 504 ) 
We now regard a plane V 0 as complex plane of the quantity x 
and we place the complex plane V x of the quantity y = <p (as) parallel 
to F, at a distance h and in such a way, that the real axes and 
the imaginary axes are each other’s orthogonal projection. Then to 
each point x of V 0 are conjugated by means of the function y=<p (a) 
one or more points y of V x . By connecting corresponding points x 
and y by rays a congruence of rays is formed which can serve as 
the image of the function y = tp {x). 
For the case y = 9 0 (a?) = x we should obtain in this way the 
congruence of rays formed by all the normals on the planes F 0 
and V x as representative of the identity. 
If now we let the function y — <p (x) gradually arise from the 
identity, then to each stage of the generating process a definite con¬ 
gruence of rays will belong. All these congruences form together a 
complex of rays. It is clear, that the formation of the function 
y = w(x) will now be represented by this complex of rays. 
Let us first examine the complex cones of the points of V 0 . Each 
point x ~ u -j- iv of this plane is the vertex of a cone counting in 
any case the normal in .r on V 0 among its generatrices; this edge 
namely intersects the plane V x in y = u~\~iv = x. 
The section of this complex cone with V x will pass through the 
point z = x and all points representing the values taken by y n — <p n {x) 
when n increases from 0 to 1. So this section also gives us a 
representation of the generating process of <p (x). It goes without 
saying that we can continue the iteration also past y = <p (x ) and 
likewise that we can also regard negative values of n. The whole 
of the complex cone embraces in fact all functions y n =. tp n (x), where 
n varies from — ae> to -j- ao. Also the section regarded as a whole 
will contain all the values of the function y n = tp n ( x ), where x is 
constant and n varies from — ^ to -j- 00 . Each value of x possesses 
its own complex cone and therefore also its own section. We shall 
indicate this section by the orbit x ~^y n . 
We might also have indicated the increase of tp (x) by allowing 
the plane V x to grow* gradually out of \\ and that by allowing 
the distance of the planes to increase regularly from 0 to h , so that 
<P n (x) is represented in the plane V„ at a height nh above F 0 . Let 
us then suppose in each plane F„ the image y n ~ <p„ (x) belonging 
to some initial-point x = u -f iv to be constructed, then all these 
points will form in their regular* succession a twisted curve. Each 
of the qd* points x of F» gives rise to a suchlike twisted curve and 
the function y — <p (x) with its different stages of development is 
thus represented by a congruence of twisted curves. 
