106 GEOMETRICAL RESEARCHES, 



If we regard the point o ]L and its polar plane as vertex and 

 axis to homologic figures whose homological ratio is 1, and 

 that we look on a^ b , c , d , &c., as belonging to one of these 

 figures, then will a^ & 2 , c g , d 2 , &c., be their corresponding points 

 in the other figure. Similarly, by regarding o 2 and its polar 

 plane as vertex and axis to homologic figures whose homological 

 ratio is 1, and looking on a 2 , & 2 , c 2 , d^ &c., as belonging to one 

 of these figures, then will a , b , c 3 , d , &c., be their corresponding 

 points in the other figure. And it is evident we may thus pro- 

 ceed until we arrive at the final extremities of the inscribed 

 n'gons. But, as homologic figures are homographic, and that 

 figures homographic with any figure are homographic with each 

 other, it is evident the first extremities of the inscribed n'gons 

 belong to a figure which is homographic with a figure to which 

 the final extremities of these ft'gons belong. Moreover, it is 

 evident that the first and last extremities of each %'gon are 

 corresponding points in the homographic figures, and that the 

 tangent planes at the extremities of each w'gon are correspond- 

 ing planes. 



Hence we may announce the following theorem : 



THEOREM 1. 



If in a surface of the second degree there be inscribed n'gons 

 such that the n successive sides of each pass in order through a series 

 of n fixed points ; then will the first extremities of these argons belong 

 to a figure which is homographic with a figure to which their final 

 extremities belong ; moreover, the extremities of each n'gon will be 

 corresponding points in the homographic figures, and the tangent 

 planes at these extremities will be corresponding planes. 



2. The two following theorems are immediate consequences 

 from theorem 1 : 



