( 789 ) 
is uniform convergence *) by the preceding ones as well as by the 
succeeding ones. | 
In the transformed figure to this answers a system of single closed 
curves *) k’, which, lying inside each other, contract from XK?’ to H’, 
whilst there is again uniform convergence to each of the curves 4’ 
by the preceding ones as well as by the succeeding ones. 
Let us now regard, starting from A and stopping in H, each curve 
k, together with the corresponding curve 4’,; here « represents the 
radius of the circle £,. 
We shall assume that the transformation in the surface leaves the 
indicatrix invariant; then in & and &’ opposite senses of circuit 
correspond to each other. 
For values of « under a certain limit @, the curve & lies inside 
k’; beyond a certain value «, however the curve 4’ lies continually 
inside &. Between «, and «, there can be values for which the inner 
domains of & and #’ lie entirely outside each other; these then have 
an upper limit «,, situated between a, and ae,. | 
Between a certain nethermost limit «@, (which with a, existing 
coincides with «,, otherwise with «,) and an upper limit «,, the 
curves / and &’ must continually penetrate each other. 
If ¢, and «, coincide, k,, and k’,, do so too, and, opposite senses 
of circuit corresponding for the transformation on this curve, we 
find two points invariant for the transformaton. 
We now assume that a, and «, do not coincide. Then &,, and 
k’,, touch each other in one or more points or ares, without pene- 
trating each other, whilst for the rest they lie either outside each 
other, or £,, lies entirely inside 4’. In the former case one domain 
is determined lying outside &,, and inside %’,,,in the latter case one 
or more domains. 
We can now choose among the domains determined by 4, and 
k', in the following way for a certain segment of values a, starting 
at a, and stopping at «ap, every time a domain y, lying outside 
k, and inside ’’,, in such a way, that to the boundary of each yz 
inside there is uniform convergence by the boundaries of the succeeding 
ys, so that the domain y, continually contracts until at ay it vanishes 
and its boundary is reduced to a point or are of single curve. 
To this end we choose in y,, an arbitrary point, and fartheron 
we choose y, in such a way that this point lies inside it; this will 
be possible up to a certain value a’; we then stop at '/, (a, + a’) =a". 
Next we choose inside the domain y,”, determined in this way, an 
1) ScHOENFLIES, Jahresber. d. D. M. V. XV, p. 560. 
2) id., Mathem. Ann. 62, p. 305. 
