383 



4m sides. A very elementary example is furnished by an in- 

 scribed plane quadrilateral, of which the two points of meet- 

 ing of opposite sides are well known to be conjugate, relatively 

 to the conic or to the surface, and are adapted to be the points 

 of meeting of opposite sides of infinitely many other inscribed 

 quadrilaterals. 



When all the sides but one, of an inscribed gauche poly- 

 gon, pass through given points, the remaining side may be 

 said generally to be doubly tangent to a real or imaginary sur- 

 face of the fourth order, which separates itself into <ii?o real 

 or imaginary surfaces of the second order, having real or ima- 

 ginary double contact with the original surface of the second 

 order, and with each other. If the original surface be an 

 ellipsoid (e), and if the number of sides of the inscribed po- 

 lygon, pp, . . . 7-2m, be odd, = 2/« + 1, so that the number of 

 fixed points Aj, . . . K^m is even, = 2m, then the two surfaces 

 enveloped by the last side Pgm f are a real inscribed ellipsoid 

 (e'), and a real exscribed hyperboloid of two sheets (e") ; and 

 these three surfaces (e) (e') (e") touch each other at the two 

 real points b, b', which are the first corners of two inscribed 

 polygons BBi . . . B2m-i ^nd b'b'i . . . B'gm-u whose 2m sides pass 

 respectively through the 2m given points (a). If these three 

 surfaces of the second order be cut by any three planes pa- 

 rallel to either of the two common tangent planes at b and b', 

 the sections are three similar and sitnilarly placed ellipses; 

 thus b and b' are two of the four umbilics of the ellipsoid (e'), 

 and also of the hyperboloid (e"), when the original surface e 

 is a sphere. The closing chords Pgm f touch a series of real 

 curves (c) on (e'), and also another series of real curves (c") 

 on (e"), which curves are the aretes de rebroussement of two 

 series of developable surfaces, (d') and (d"), into which latter 

 surfaces the closing chords arrange themselves ; but these two 

 sets of developable surfaces are not generally rectangular to 

 each other, and consequently the closing chords themselves 

 are not generally perpendicular to any one common surface. 



