360 
modifieation, can be followed in the manner described above by a 
continuous modification of the representation of the residual domains 
of d., determined by @,, furnishing us with a representation a, dis- 
tinguishing itself thereby from «, that a domain of the second and 
one of the third kind have been united into a single domain of the 
first kind; this domain however, if it does not oceupy the whole 
sphere u, can be absorbed in the manner described above by an 
adjacent domain of the second or of the third kind, by which process 
a, passes continuously into a representation «,, distinguishing itself 
, that a domain of the second and one of the third 
kind have been absorbed together by a domain of the second resp. 
of the third kind. 
By repeating this operation as many times as possible we arrive 
after a finite number of steps at a representation @=) for which the 
thereby from « 
domains d, are either all of the second or all of the third kind. So 
this representation is a canonical one, and we have proved: 
Theorem 2. All univalent continuous transformations of the same 
degree of a sphere in itself belong to the same class. 
A proof of the inverse theorem has been given Mathem. Ann. 71, 
p. 105. 
In carrying out the ideas sketched in the second communication 
on this subject’) | experienced that in some points of the course of 
demonstration indicated there, still a tacit part is played by the 
Schoentliesian theory of domain boundaries criticized by me”), 
so that the theorems 1 and 2 formulated p. 295 and likewise the 
“general translation theorem’ founded upon them and enunciated 
without proof Mathem. Ann. 69, p. 178 and 179, cannot be considered 
as proved*), and a question of the highest importance is still to be 
decided here. | 
The “plane translation theorem” stated at the end of the second 
communication (p. 297) and likewise Mathem. Ann. 69, p. 179 and 
180, has meanwhile been proved rigorously by an other method.*) 
1) These Proceedings XIL (1909), p. 286— 297, 
2) Compare Mathem. Ann. 68 (1910), p. 422—434. 
3) Already the property of p 288 that the transformation domain constructed 
jn the way indicated there determines at most two residual domains, vanishes for 
some domains incompatible with the Schoenfliesian theory. 
t) Compare Mathem. Ann. 72 (1912), p. 37—54. 
