285 ) 



Fig. 1. 



XXIV. — On FagnmiVs Theorem. By H. F. Talbot. 



(Read 20th April 1863.) 



Before proceeding to the subject of this paper, I wish to advert to the follow- 

 ing well-known theorem. 



Let ABC be any triangle with its inscribed and circumscribed circles. Then 

 if we take any other point D in the exterior circle, 

 and draw the tangents DE, EF, FD, the last tangent 

 will come again to the original point D. A similar 

 theorem is true of a polygon of any number of sides 

 which is both inscribed in, and circumscribed to, a 

 circle. Also if ellipses are substituted for circles. 

 Of these theorems I remember to have seen a de- 

 monstration founded on the theory of elliptic in- 

 tegrals. But this appears to me to be a great 

 waste of analytic power; for the theorem really 

 results from first principles, as I think will be 

 manifest from the following considerations. 



I shall take the simplest case, that of two circles inscribed in, and circum- 

 scribing, a triangle, because the reasoning is similar in the more complicated cases. 



Lemma 1. If there are two circles, one within the other, 

 and if AB, CD, are two chords of the outer circle, touching 

 the inner circle, then if C lies between A and B, D does not. 

 For it is plain that a line drawn from C to any other 

 point in the arc AB, would not touch the inner circle, which 

 is contrary to the hypothesis. 



If a circle lies anyhow within another, and tangents 



1, 2—2, 3—3, 4, be drawn, 

 then, generally speaking, the points 1 and 4 do not 

 coincide. The arc 1, 4, may be called the deviation, 

 and if more tangents 4, 5 — 5, 6 — 6, 7, be drawn, 

 the arc 4, 7, is the second deviation. 



Lemma 2. The successive deviations are con- 

 stantly in the same direction, and consequently the 

 total deviation increases till at length it surpasses 

 any given arc of the circle. 



Demonstration. Since the point 4 (as repre- 

 sented in fig. 3) lies between 1 and 2, the point 5, 

 at the other end of the tangent, lies between 2 and 

 3 (by the first lemma), and for the same reason the point 6 lies between 3 and 4. 



VOL. XXIII. PART II. 4 I 



