ON THE POINCARE TRANSFORMATION. 443 



arc of the principal braucli the rotation being in the negative sense. Take for 

 second "leg" of the normal contour the segment AjB-, = S2 directed inward 

 and in which Bo is the first point of zero deviation encountered. B2 may be 

 a point on the principal branch (Fig. 2) or on the secondary branch (Fig. 3). 

 Selecting then for third leg the normal arc a^ = B2A3 and continuing in this 

 manner Ave shall finish by returning to the point Bi, after having described a 

 closed contour (K) everywhere convex and consisting of normal arcs and seg- 

 ments only. This contour is shown in the figures by the heavy lines; its image 

 by heavy dotted lines. 



•5. Proof of Poincare's Theorem. 



If ai is the image of the normal arc ai it is clear that §,1 cannot intersect (K) 

 m any other part of it but the corresponding arc ai, for a.i and ai are contained 

 between the sane two rays li and li = 1. On the other hand if Sk is the image of 

 the segment Sj^, then Sk will have no other points in common with the con- 

 tour (K) than the point Bk. ^ 



From these remarks the proof of the theorem follows without difficulty. 

 For if we assume that there are no invariant points, no arc ai would have any 

 points in common with the corresponding arc ai. The contour (K) would, 

 therefore, be either entirely within or entirely without its imago (K) and in 

 either case the area of the ring (Kc) could not equal that of ring (Kc) con- 

 trary to the hypothesis of conservation of areas. 



*In the above nientioned article Poincare states the theorem in the case of con- 

 centric circles. Birkhoff also considers this case, although he remarks at the end ot 

 his article that the theorem could be extended to the case of any two convex contours 

 with the aid of a conformal transformation. This has never been very clear to me. 



