475 



sketch, and on which he hopes to return in a future commu- 

 nication. 



Theorem. — Let A u A 2 , . . . A n be any n points (in num- 

 ber odd or even) assumed at pleasure on the n successive sides 

 of a closed polygon BB y B 2 ... B n . x (plane or gauche), in- 

 scribed in any given surface of the second order. Take any 

 three points, P, Q, R, on that surface, as initial points, and 

 draw from each a system of n successive chords, passing in 

 order through the n assumed points {A), and terminating in 

 three other superficial and final points, P', Q, R'. Then there 

 will be (in general) another inscribed and closed polygon, 

 CCi C 2 ... C n . i, of which the n sides shall pass successively, 

 in the same order, through the same n points {A) : and of 

 which the initial point C shall also be connected with the 

 point B of the former polygon, by the relations 



ael |3y del /3'y' 



where 





be 



az\ be aYA' 









bfin 



ya b'f'm' yd 









ca 



pZ/x ~ c'd fiZ'n" 









cgn 

 ab 



a/3 c'g'ri a'/3' 

 yrjv a'b' y'rjV 







a - 



-QR, 



b = RP, 



c 



= PQ, 



e = 



-BP, 



f=BQ, 



9 



= BR, 



I - 



-CP, 



m= CQ, 



n ■■ 



= CR, 



a '■- 



- QR', 



V = R'P', 



c - 



- P'Q, 



e'- 



-BP, 



f=BQ, 



9' 



= BR', 



r.. 



-CP, 



m'= CQ, 



n' 



= CR'; 



while aj3y£^r?Xjuv, and d^'y'^XnXfjLv, denote the semidiame- 

 ters of the surface, respectively parallel to the chords abcefglmn, 

 db'c'e'f'g'l'm'n'. 



As a very particular case of this theorem, we may sup- 

 pose that PQRPQR' is a plane hexagon in a conic, and BC 

 its Pascal's line. 



