442 PROCEEDINGS OF THE INDIANA ACADEMY OF SCIENCE. 



I shall call the curve (d) deprived of all loops the principal branch of 

 the locus (D). If (D) possesses more than one such branch, the one "closest" 

 to the inner Iwundary may be selected for the principal branch. 



' .1 I'drliriilnr ('ni^e 



I will say that a closed contour (K) is everywhere corivex if any niy thru () 

 /ueete it in one and only one point (Fig. I). If a contour (K) is not every- 

 where convex, it is clear that there exist rays which touch it. By drawing all 

 these tangent rays it is possible to divide the contour into "convex" and 

 concave" ai'cs and there is a finite number of these arcs. (Fig. 2). 



It 13 e\Tdenl from the foregoing considerations that if the principal branch 

 IS everywhere convex, this is also true of its image (d). In the general case 

 by drawing the tangent ray^ we simultaneously divide both (d) and (d) into 

 r-onvex and concave arcs. 



TIk'sc preliminaries being established, the proof of the theorem is im- 

 mediate in tie case when the principal branch of the zero deviation curie is 

 everywhere convtx. Indeed (d) and (d) must in this case have at least two real 

 intersections, for othenvise d would he either entirely within (d) or entirely 

 without In either case, the area of the ring (d, c) could not equal that of the 

 rinff (d, c) contrary to the hypothesis of conservation of areas. If now I is 

 a point common to (d) and (d), its image I coincides with I, and the proposi- 

 tion is proved. 



The method'used here to prove that (d) and (d) intersect in at least two 

 points, applies to the general case and discloses this fundamental fact: // 

 the point I is situated on a convex arc of the principal branch it is certainly an 

 iniarwnl point. If, however, the point I is on a concave arc it may not be 

 ati invariant point, as for instance the point C in Fig. 2. The problem, there- 

 fore, reduces to showing that at hast one convex arc of the branch (d) meets 

 itif xmage 



7. The Auxiliary Contour. 



I shall call an arc of zero deviation a normal arc if it is possible to go from 

 one extremity of the arc to the other without changing the sense of rotation. 

 A segment of a ray thru O is normal if it is j)ossible to go from one extremity 

 to the other without changing the sign of the de\iation. A contour consist- 

 ing of normal arcs and segments, I shall call a normal contour. 



Lemma B. It is always possible to construct within the ring (Cc) a closed 

 normal contour iK) completely surrounding the boundary (c) and everywhere convex. 



I comrience by drawing all the rays taiLgent to the zero deviation curve 

 both in its principal and secondary brr-r.ehes. The ?oeus (D) as well as its 

 image (D) is thus divided in a certain number of cc^jivex and concave arcs 

 (Figs. 2 and 3). Any one of these tangent rays li touches (D) ir Ai and crossos 

 it besides in a number of points B|, B'l; Let ai = BjAj be a normal 



