the In-and-circumscribed Polygon. 



343 



Wc have, therefore, 



A=l 



2B = M + 2 

 -8C = M2-4M 

 16D = M3-2M2 

 -128E = 5M'*-8M3 

 &c. 

 1024(CE-D'2) = M'»(M^-12M + 16) 

 &c. 

 Hence for the ti-iangle, quadrangle and pentagon, the condi- 

 tions are — 



I. For the triangle, 



M + 2=0. 



II. For the quadrangle, 



M-4 = 0. 



III. For the pentagon, 



M'^-12M + 16 = 0. 



And so on. 



It is worth noticing, that, in the case of two conies having a 

 4-point contact, we have F=0, and consequently M = l. The 

 discriminant is therefore (1+^)^, and as this does not contain 

 any variable parameter, the conies cannot be determined so that 

 there may be for a given value of n (nor, indeed, for any value 

 whatever of n) an infinity of ?i-gons inscribed in the one conic, 

 and circumscribed about the other conic. 



The geometrical properties of a triangle, &c. inscribed in a 

 conic and circumsci'ibed about another conic, these two conies 

 having double contact with each other, are at once obtained from 

 those of the system in which the two conies are replaced by con- 

 centric circles. Thus, in the case of a triangle, if ABC be the 



triangle, and «/3y be the points of contact of the sides witli tht 



