In both cases 0 determines on the sphere two regions, V and W, 
in one of which lies its image O'-, then in the other lies O'i, i.e. its 
image for the inverse transformation. If we now repeat the trans¬ 
formation itself as well as its inverse an indefinite number of times, 
we obtain on one hand in V a series of domains O', O', O'",..., and 
on the other hand in IT a series of domains Oi,0"i, O n i,... 
In neither of these series exist intermediary regions between the 
successive terms, and each series converges to an invariant limit set, 
which causes the existence of at least one invariant point. These in¬ 
variant limit sets may cohere with each other (in the second case 
of fig. 9); then we are sure of only one invariant point, otherwise 
always of two. 
If in the Cartesian plane we have a transformation domain breaking 
its connection, either one of the cases of fig. 9 appears again, or the 
domain is bounded by an arc of curve running from infinite to infinite, 
or by two such arcs of curve. 
If it is bounded by one arc, the existence of an invariant open 
system of curves can be deduced according to the proof of theorem 1. 
If it is bounded by two such arcs B , and B„ we can, if no 
Fig. 10 
