936 



We shall now understand by t the transformation carrying each 

 point P of r on the line of ^ passing through it along an arc of 

 constant length /; to the left, if P lies in (/i or on the boundary 

 of (/i ; upward, if P lies in //, ; to the rlyht, if P lies in y, or on 

 the boundary of g^. This transformation t is duly biuniform and 

 continuous, lias no invariant point, and leaves the indicatrix invariant. 

 But if for each positive or negative integer n including zero we 

 represent the image of /'' for the transformation ^" by „y^and5(„P) 



n 



by P' , P' does not depend on P continuously (for, if the sequence 

 of points P^, P^, Pt, . . . lying in 7, converges to the point Plying 

 on the line y = — i, the sequence P/, P/, P/, . . . . does not converge 

 exclusively to P' , but also to other points of T). 



IVius neither can F be represented hiuni/ormlii and continuously 

 on a Cartesian plane C in such a way that the image of the trans- 

 formation t of r is a translation of C. For, if such a representation 

 were possible, P' would necessarily depend on P continuously. 



