90 GEOMETETCAL RESEARCHES, 



successive sides of each pass in order through n fixed points o v o 2 , 

 .... o n ; then, d l d 2 .... d.^ representing an inscribed rigon ivhose 

 sides pass in like manner through the fixed points., and which 

 we may conceive to be deformed so as to assume the 

 position of all the rigons which can be so inscribed., we will have 



3. We know that the first extremities of the inscribed n'gons 

 are corresponding points to their last extremities in a pair of 

 homographic figures. We know also that according as these 

 homographic figures are homologic or not homologic so according- 

 ly will the closing chords of the w'gons all pass through one point 

 or be tangents to a conic T having double contact with S, (in the 

 points which are answerable positions for first extremities of 

 inscribable closed w'gons whose sides pass in order through the 

 given points) and of like or unlike rotatives, (reckoning from the 

 final extremities of the n'gons) just according as the final ex- 

 tremities of the ^'gons are on the same side or on opposite sides 

 of the line of contact of the conies T and S. But, when we can 

 interchange the distinct extremities of one of the inscribable 

 w'gons, we know that the figures are homologic. Hence we 

 have the following theorems : 



THEOREM 2. 



If in a curve of the second degree there can be one inscribed 

 closed 2 rtgon ivhose two successive series of n sides pass in order 

 through n. fixed points, and are not co-incident ; then will any point 

 in the curve be an answerable position for the first extremity of 

 another inscribable closed 2 rigon whose sides will pass in like order 

 through the *& fixed points. 



And the closing chords of the inscribable rigons the sides of each 

 of which pass in order through the n fixed points will all pass 

 through one point, whose polar cuts the curve in the answerable 

 positions for first extremities of inscribable closed rigons whose sides 

 pass in order through the points. 



