the in-and-circumscribed Polygon. 377 



The second theorem gives the condition in the case where 

 the conies are replaced by the circles aP + y 2 — R 2 = and 

 (#— «) 2 + «/ 2 — r 2 =0, the discriminant being in this case 



As a very simple example, suppose that the circles are concen- 

 tric, or assume « = 0; the square root of the discriminant is here 



(l+£)vWR 2 f; 



R 2 



and putting for shortness — ^ —ex., we may write 



A + Bf-f ...=(l + f)/l + «f, 

 t. e. 



E =-A" 4+ r6 a3 ' &c - 



thus in the case of the pentagon, 



CE_I)2== li" 4 {(«~*)(6«-S)-4(«-2)«} 



= I 4l a V-12«+16), 



and the required condition therefore is 

 a 2 -12a + 16=0. 

 It is clear that, in the case in question, 



r QAO ^5 + 1 



_ = cos3 6= -— ; 



2. e. - = 4/5-I, or (R + r) 2 -5r 2 =0, viz. ( ^a + l) 2 -5 = 0, 



i. e. a + 2 */u— 4=0, the rational form of which is 



a 2 -12« + 16=0, 



and we have thus a verification of the theorem for this particular 

 case. 



2 Stone Buildings, Oct. 10, 1853. 



Phil Mag. S. 4. Vol. 6. No. 40. Nov. 1853. 2 C 



