Mathematics. — "Remark on the plane translation theorem". By 

 Prof. L. E. J. Bhouaver. 



(Communicated in the meeting uf December 28, 1918). 



The plane translation theorem enunciated by me in Vol. XII of 

 tliese Proceedings (p. 297) and completely proved for the first time 

 in Vol. 72 of the Mathematische Annalen (p. 37 — 54), runs as follows : 



A continuous one-one transformation of the Cartesian plane F in 

 itself with invariant indicatrix, but nnthout an invariant point, is a 

 translation all over the plane. 



W^e mean by this that each point of V is situated in a translation 

 field, i. e. in a region lying outside its image region and bounded 

 by two simple open lines not meeting each other, one of which is 

 the image of the other. 



Let t be the given transformation, T a translation field belonging 

 to t, nT for each positive or negative integer n including zero the 

 image of T for the transformation t". The set of points T' = '^ (nT) 



n 



can be represented binniformly and continuously on a Cartesian 

 plane C in such a way that the image of the transformation t of 

 T' is a translation of C. Thus, if by a convenient choice of T we 

 can arrange T' to fill up the lohole plane r, r can be represented 

 biuniformly and continuously on a Cartesian plane C in such a 

 way that the image of the transfornmtion t of F is a translation of C 



However the question, ivJiether for each transformation t a choice of 

 T making T' to fill up the lohole plane r, is possible, must be 

 answered in the negative, as was indicated by me in a footnote 

 on page 37 of the quoted paper of the Mathematische Annalen, 

 and as appears from the following example : 



In r we define a Euclidean system of measurement, and a rec- 

 tangular system of coordinates founded on it. The straight lines 

 y = 1 and y = — 1 divide rinto three legions^^, Cv>l),^, (1>y >— 1). 

 and g,{y<i — i). Each of the regions g, and (/, we fill up with a 

 pencil of lines y = c, and the region g, we fill up with a pencil 



of lines if = ~ . These three pencils together with the lines 



1 -\-,ü — c 



y := 1 and y =i —1 form a pencil /i of simple open lines not inter- 

 secting each other, and covering F entirely. 



