56 



MR TALBOT ON CONFOCAL CONIC SECTIONS. 



scribed to the other, of two confocal ellipses, its perimeter is constant, at what- 

 ever point of the exterior ellipse it is supposed to commence. 



But the truth of this can be shown without any reference to Graves's theorem, 

 from the simple consideration that two consecutive sides of the triangle make 

 equal angles with the periphery of the exterior ellipse. Hence if the point of 

 departure, or vertex of the triangle, suffers a very small displacement, the three 

 sides increase or diminish at one end by three small quantities 8, 8', 8" (generally 

 speaking all different). 



Let us suppose this to occur at the right extremity of each of the three lines, 

 then it is evident that the increments (or decrements as the case may be) which 

 occur at their left extremities will be — 8", —8,-8' respectively (because each 

 side gains at one end what the following side loses there). Therefore the total 

 increase of the perimeter = (8 — 8") + {8'— 8) + (8"— 8') = 0. A much more 

 general theorem can be proved in the same way. " If a triangle cirumscribes an 

 ellipse, and its three angles rest upon the peripheries of three other ellipses (all 

 four having the same foci), the perimeter of the triangle is constant." 



I find that Chasles has given this theorem (although without proof) in his 

 memoir, which I have already quoted (see my last paper, p. 287). The same is 

 true of polygons of n angles resting upon n confocal ellipses. 



I will conclude this short note by giving a curious property of the circle, com- 

 municated to me by C. H. Talbot, Esq. 



" If three concentric circles (fig. 3) are described from any centre S, with 



■H 



radii m, m + h, m + 2h. And if three other concentric circles intersecting them 

 are described from any other centre H, with radii n, n + k, n + 2h [m, n, h having 



