MR TALBOT ON FAGNANI S THEOREM. 



287 



From this theorem a very remarkable consequence may be drawn. Let ABC, 

 DEF, be the two ellipses having same centre O and foci S, H. Moreover, let the 

 axis major of one bear such a proportion to that of the other, that if two chords, 

 AB, BC, be drawn in the outer 

 ellipse touching the inner one at 

 E and F, the third chord CA 

 shall also touch it at D. Then 

 by Graves's theorem BE + BF— 

 arc EF = C, a constant. Also 

 CF + CD-arc FD = C, and AD + 

 AE - arc DE = C. Therefore add- 

 ing these equations together, we 

 find that the periphery of the 

 triangle ABC, minus the peri- 

 phery of the ellipse =3 C. Now, 

 by the preceding theorem I., we 

 know that if, instead of iV, we take 

 any other point A' of the outer Fig. 4. 



ellipse as the origin, and draw tangents A'B', B'C, CA', from it to the inner 

 ellipse, they will form a closed triangle. Its periphery, minus that of the ellipse, 

 will = 3 C, by the same reasoning as before. Therefore the peripheries of the two 

 triangles are equal. But the same reasoning applies to a closed polygon of any 

 number of sides. Whence the following theorem : — " If two ellipses, with same 

 centre and foci are such that a polygon can be inscribed to the one and circum- 

 scribed to the other, the periphery of the polygon is constant, whatever point of 

 the outer ellipse be assumed as its origin."* 



After I had arrived at this theorem, I found that M. Chasles had given one 

 nearly resembling it (Comptes Rendus, 1843, p. 838), with the additional remark, 

 that since every two consecutive sides of the polygon are equally inclined to the 

 periphery of the outer ellipse, a ray of light might describe such a polygon, by 

 successive reflections at the periphery ; and therefore it would always be a 

 tangent to an inner confocal ellipse. But I think this remark may be rendered 

 yet more striking by considering the cases in which the polygon is not closed. If 

 this occurs when the origin is any point A, it will also occur when any other 

 point is assumed as the origin. The path of the ray of light under these circum- 

 stances would be constantly varying, but always tangent to the inner ellipse, so 

 that ultimately, after an infinite number of reflexions, the ray will have passed 



* It is unnecessary to say with Chasles that they must be polygons of the same number of 

 sides, for they cannot be otherwise. M. Chasles, in his paper here referred to, gives no demonstra- 

 tions. I am not aware whether they were subsequently published. 



