BY MAETIN GARDINER, C.E. 101 



answerable position for p l . Hence we re-arrive at theorem 5, 

 and also at the following : 



THEOREM 13. 



If ^ a 2 a n a i 



inscribed in a circle, the sides of each of which pass in order through 

 n fixed points o x , o 2 , o n , and that 



a o 



n n 



when the rotatives of the involved lines are taken in respect to any 

 point in the circumference; then will any point in the circumference 

 be an answerable position for the first angular point of an inscrib- 

 able closed n'gon whose sides pass in order through the n fixed 



points. 



12. When the data is such as to render the problem poris- 

 matic when the number of points is even, and that they are all 

 in one straight line ; then, by supposing the points a^ and b l 

 co-incident with the points in which this line cuts the circle, it is 

 obvious from the last theorem that 



which is a formula already arrived at by Chasles, on page 465 of 

 his treatise on " Geonietrie Superieure" 



13. Again (since the extremities of all the inscribable ^'gons 

 belong to homographic figures) the following theorem can be 

 easily deduced : 



THEOREM 14. 



If there be given a circle and n points in a plane, two straight 

 lines, X and Y can be found, such that if a^ and a be the ex- 

 tremities of any inscribed n'gon ivhose sides pass in order through 



