( 723 ) 
the rectangle A x B x S R a tangent curve «_3 , joining two points 
A 3^ and _Z>£ of y and tf between rand« x . The rectangle A 9 B 0 B x A x 
is then divided into four regions. In these we choose in the way 
described above successively the points Q\_, Qi, Q±, Ql, draw 
through Q\_ a tangent curve at joining two points A 1 and B± of y 
and tf, and we deal analogously with the other three points. 
Going on in this manner we construct for each fraction —■ <? 1 a 
2" ^ 
tangent curve «« joining two points of y and tf; two of these curves 
2 n 
chosen arbitrarily can coincide partially, but they cannot cross each 
other. 
All these tangent curves must now cover everywhere densely the 
inner domain of the rectangle A # B 0 B l A x . For, if they left there 
open a domain G, then a domain G' n bounded by two tangent curves 
with indices -- and . —~ would converge to G. For n sufficiently 
2» 2 n 
great however the point Q-ia+i would then lie inside G, thus in 
contradiction to the supposition also a tangent curve would 
pass through G. 
From this ensues, that, if we add ( the limit elements of the tangent 
curves a a , which are likewise tangent curves, the inner domain of 
the rectangle A 0 B„ B y A 1 is entirely covered, and further there is 
for each real number between 0 and J one and not more than one 
of these tangent curves having that number as its index. 
The structure of the field in the vicinity of an isolated 
singular point. First principal case. 
We surround the point zero P, supposed isolated, with a simple 
closed curve c, inside which lies no further point zero. And we 
assume as a first principal case that c can be chosen in such away 
that inside c no simple closed tangent curve exists, inside which P 
lies. On account of the first communication there can exist inside c 
neither a simple closed tangent curve, outside which Plies. We now 
distinguish 2 cases: 
