385 



being obliged to pass through fixed points. In this problem 

 the last side may be said to touch two imaginary surfaces of 

 the second order, which intersect each other in two real or 

 imaginary conies, situated in two real planes ; and when these 

 two conies are real, they touch the original ellipsoid in two 

 real and common points, which are the two positions of the 

 first corner of an inscribed polygon, whose sides pass through 

 the 2??z - 1 fixed points. Every rectilinear tangent to either 

 conic is a closing chord Pom-i p ; but no position of that clos- 

 ing chord, which is not thus a tangent to one or other of these 

 conies, is intersected anywhere by any infinitely near chord 

 of the system. These results were illustrated by an example, 

 in which there were three given points ; one conic was the 

 known envelope of the fourth side of a plane inscribed qua- 

 drilateral ; and this was found to be the ellipse de gorge of a 

 certain single-sheeted hyperboloid, a certain section of which 

 hyperboloid, by a plane perpendicular to the plane of the el- 

 lipse, gave the hyperbola which was, in this example, the other 

 real conic, and was thus situated in a plane perpendicular to the 

 plane of the ellipse. And to illustrate the imaginai'y charac- 

 ter of the enveloped surfaces, or the general non-intersection 

 (in this example) of infinitely near positions of the closing 

 chords in space, one such chord was selected ; and it was 

 shown that all the infinitely near chords, which made with this 

 chord equal and infinitesimal angles, were generatrices (of 

 one common system) of an infinitely thin and single-sheeted 

 hyperboloid. 



Conceive that any rectilinear polygon of n sides, bBj . . ." 

 B„_i, has been inscribed in any surface of the second order, 

 and that n points Aj . . . a„ have been assumed on its n sides, 

 BBi, . . .B„.iB. Take then at pleasure any point p upon the 

 same surface, and draw the chords pA]Pi, . . . Pn-i a„p„, passing 

 respectively through the n points (a). Again begin with p,„ 

 and draw, through the same n points (a), n other successive 



