490 a “ smith’s prize ” paper ; solutions. [534 
The porism in regard to the two conics is, that in general it is not possible to 
find any polygon of n sides satisfying the conditions; but that the conics may be 
such that there exists an infinity of polygons; viz. any point whatever of the one 
conic may then be taken as a vertex of the polygon, and then constructing the figure, 
the (n + l) th vertex will coincide with the first vertex, and there will be a polygon of 
n sides. 
Now imagine that the conic touched by the sides is a circle having double contact 
with the other conic. Describe any one of the polygons, and with each vertex as 
centre describe the orthotomic circle, which will, it is clear, be a circle passing 
through the points of contact with the fixed circle of the sides through the vertex. 
We have thus a closed series of n circles, each touching the two adjacent circles of 
the series. And by considering any other polygon, we have a like series of n circles: 
and by what precedes the envelope of all the circles of the several series is a pair of 
circles; that is, the circles of every series touch these two circles. We have consequently 
two circles, such that there exists an infinity of closed series of n circles, each circle 
touching the two fixed circles, and also the two adjacent circles of the series; which 
is the porism arising out of the second problem.