THE IN-AND-CIECUMSCEIBED POLTaOjS^ 
237 
It wiU be remarked that [41=^( — n — 1) breaks up into the factors ^ and v-\-l ; and so 
[6]=(2^ — V — l){4vi^-}"(‘'4"l)^} breaks up into the factors 2^ — v — 1 and 48'f+({'+l}^ 
It may be added that the developed expression of [6] is 
= r ^^8v 
I +i.2(»+l)’ 
SO that the difference between this and [5] is ^^.4(^-}-l)^ which is =f7^; this agrees with 
a former result. 
M. Mention has also given, but not in a developed form, the formulae for the enneagon 
and the endecagon, and the following formula for the decagon, viz. 
IV. 
Considerations as fo the form of relation^ in the case of tioo circles, for Polygons of cm 
odd and even number of sides resggectively. 
The relation between the two conics, or condition for the existence of the polygon, is 
the same whatever point of the circumscribed conic is taken as an angle of the polj^gon. 
Take for an angle, a point of intersection of the two conics. Consider first the case 
of the triangle ; if a point of intersection is taken as a vertex A of the triangle, then 
the sides AB, AC coincide in direction with the tangent at A to the inscribed conic U, 
hence B and C coincide together at the point where this tangent meets the circumscribed 
conic V, BC is therefore a tangent of V, but it is by hypothesis a tangent of U ; hence 
for the triangle the relation between the inscribed conic U and the circumscribed conic 
V is as follows, "viz. a tangent to U at a point of intersection with V meets V at a point of 
contact of a common tangent of U and V. 
In like manner for the quackangle, if A be taken at a point of intersection, the sides 
AB and AD will coincide in direction with the tangent to U at this point, consequently 
B and D must coincide at the point where this tangent meets V ; hence also CB, CD, 
the two tangents to U from the point C, must coincide, or C must be a point of inter- 
section of the conics U, V. In other words, the pole, with respect to the inscribed 
conic U, of a common chord AC of the two conics must lie on the circumscribed conic 
V; this is therefore the condition for the quadrilateral. 
In the ordinary mode of drawing the figures, with two conics which do not intersect, 
the points and lines employed in the foregoing constructions are imaginary, but the 
conics may be so drawn that these points and lines are all real. 
In general, for a polygon of an odd number, 271+1, of sides, then starting from a 
point of intersection, the sides will coincide in pairs, viz. the first and last, second and 
last but one, and so on, the middle or (w+l)th side being a common tangent of the two 
