440 PROCEEDINGS OF THE INDIANA ACADEMY OF SCIENCE. 



On the Poincare Transformation. 



Tobias Dantzig. 

 1. Introductory. 



In a memior entitled "Sur iin Theoreme de Geometrie" (Rendieonti del 

 Circolo Matematieo di patermo, Vol. 33, 1912, P. 375-407) the late Henri 

 Poincare has considered a certain type of transformations of fmidamental 

 value in Celestial Mechanics. Without giving a proof he has announced 

 there a general property of all such transformations. The proposition has 

 since been taken up by George D. Birkhoff who in his paper "Proof of 

 Poineare's Geometrie theorem" Transaction of the American Mathemati- 

 cal Society, Vol. 14, 1913) has given the theorem a general demonstration. 

 His proof lacks, however, simplicity and directness. 



In my article entitled "Demcmst ration directe du dernier theoreme de 

 Henri Poincare" which appeared in the February issue of the "Bulletin des 

 Sciences Mathematiqucs et Aslronomiques." I gave an elementary, genetic 

 proof of the j)rf)position. 1 wish to reproduce here the main features of my 

 demonstration as well as to bring out in greater detail some points which were 

 left incomplete in the said i)a])er. 



2. Poincnre's Theorem. 



Slightly generalized* the theorem can be stated thus: 



Lei T be <i Irnnsfontialion operalittq in a pUuie and hariiig the following 

 properties: 



(a) It is continuous and one-to-one in the ring formed by two closed curves 

 contours (C) and (c) of ivhich (c) is entirely within (C). {Fig. 1.) 



(b) It leaves the two contours (C) and (c) invariant. 



(c) It moves any point M oji (C) into a point M in the positive sense of 

 rotation, while the points ni on (c) advance in. the opposite sense. 



(d) It takes every point P tvithin the ring (Cc) into a point P also within the 

 ring. 



{e) It conserves areas. 



Under these considerations there are within the ring (Cc) at leasi two points 

 I and J which are left invariant by T. 



,i. Xotations. 



Choose at random within (e) (Fig. 1) a point O, and a half-line OX, for 

 poI« and polar axis respectively, and let 



,|f.f(r.O, 



^^^\e = g (r. e) 



be the polar equations of the transformation, r, O; r, are the co-ordinates 



