the In-and-circumsa'ibed Polygon. 341 



Whence, also, 

 ?''n024(CE-D2) = 



{5(R2-«2)4_8(R2_rt2)2(R2 + 2/-2);-2+16«V} { (R2-a2)2_4RV) } 



which after all reductions is 



(R2_a2-)6 



-12R-(R2-fl2)V2 



+ 16R2(R2 + 2«2)(R2-a2)2,-4 



-64RV;-6. 

 Hence the condition that there may be inscribed in the circle 

 x'^-\-y^ — '?i^=0, and circumscribed about the circle {x — aY-\-y'^ 

 — ?-^ = 0, an infinity of n-gons is for ?i = 3, 4, 5 respectively, 

 i. e. in the case of a triangle, a quadrangle and a pentagon, is as 

 follows. 



I. For the triangle, the relation is 



(R^-a2)2_4R2^2 = 0, 

 which is the completely rationalized form (the simple power of a 

 radius being of course analytically a radical) of the well-known 

 equation 



a2=R2_2Rr, 



which expresses the relation between the radii R, r of the cir- 

 cumscribed and inscribed circles, and the distance a between 

 their centres. 



II. For the quadrangle, the relation is 



(R2-fl2)2_2r2(R2 + «2)=0, 



which may also be written 



(R + ;. + a)(R + r_«)(R_?- + a)(R-;— «)-r'» = 0. 



(Steiner, Crelle, vol. ii. p. 289.) 



III. For the pentagon, the relation is 



(R2_«2)6_12R2(R2_«2)4,.2 + 16K2(R2 + 2e2)(R2_a2)2^4 



-64RVr« = 0, 

 which may also be written 



(R2 -«2)2{(R2_ a2)2_ 4R2,.2|2_ 4R2,.2{ (R2_ ,,2)2_4«2^2p 3=0. 



The equation may therefore be considered as the completely 

 rationalized form of 



(R2_a2)3 + 2R(R2-«2)V-4R2(R2-«2)r2_8Ra2r3=0. 



This is, in fact, the form given by Fuss in his memoir " De 

 Polygonis Symmetricc irregularibus circulo simul inscriptis et 

 circuniscriptis," Nova Ada Pdrop. vol. xiii. pp. 166-189 (I quote 



