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DEVELOPMENTS ON THE PORISM OF THE 
[128 
which after all reductions is 
(R 2 — a 2 ) 6 
-12 R 2 (R 2 -a 2 ) 4 r 2 
+ 16 R 2 (R 2 + 2a 2 ) (R 2 — a 2 ) 2 r 4 
— 64-R 2 a 4 r 6 . 
Hence the condition that there may be, inscribed in the circle x 2 + y 2 — R 2 = 0 
and circumscribed about the circle' (x — a) 2 + y 2 — r 2 = 0, an infinity of w-gons, is for 
w = 3, 4, 5, i.e. in the case of a triangle, a quadrangle and a pentagon respectively, 
as follows. 
I. For the triangle, the relation is 
(R 2 - a 2 ) 2 - 4E 2 r 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 
a 2 = R 2 — 2 Rr, 
which expresses the relation between the radii R, r of the circumscribed and inscribed 
circles, and the distance a between their centres. 
II. For the quadrangle, the relation is 
(R 2 - a 2 ) 2 - 2r 2 (R 2 + a 2 ) = 0, 
which may also be written 
(R + r + a) (R + r — a) (R — r + a) (R — r — a) — r* = 0. 
(Steiner, Crelle, t. n. [1827] p. 289.) 
III. For the pentagon, the relation is 
(R 2 - a 2 ) 6 - 12R 2 (R 2 - a 2 ) 4 r 2 + 16E 2 (R 2 + 2A 2 ) (R 2 - a 2 ) 2 r 4 - 64R 2 a 4 r 6 = 0, 
which may also be written 
(R 2 - a 2 ) 2 {(R 2 - a 2 ) 2 - 4R 2 r 2 } 2 - 4R 2 r 2 {(R 2 - a 2 ) 2 - 4a 2 r 2 } 2 = 0. 
The equation may therefore be considered as the completely rationalized form of 
(R 2 — a 2 ) 3 + 2R (R 2 — a 2 ) 2 r — 4R 2 (R 2 — a 2 ) r 2 — 8Ra 2 r 3 = 0. 
This is, in fact, the form given by Fuss in his memoir “De polygonis symme- 
trice irregularibus circulo simul inscriptis et circumscriptis,” Nova Acta Petrop. t. xm. 
[1802] pp. 166—189 (I quote from Jacobi’s memoir, to be presently referred to). Fuss 
puts R + a = p, R — a = q, and he finds the equation 
pig 2 _ r 2 (^p + q 2 ) ^ j'q — r 
r 2 q 2 — p 2 (r 2 + q 2 ) ~ v q +p’