394 



Mr. M. Gardiner on Undevelopable 



[May 14, 



the same generator ; and, making them first extremities, proceed to inscribe 

 2n'gons. 



(1) If these assumed points be found to be first extremities of closed 

 w'gons, the problem is fully porismatic, and every point in the surface will 

 be the first extremity of an inscribable closed w'gon. 



(2) If two of the points be found to be first extremities of closed w'gons 

 and the other not, then the line xx through these two points, and the line 

 a reciprocal to xXy pierce the quadric in four points (the punctures made 

 by a being real or imaginary as may be) which constitute the first extremi- 

 ties of all the inscribable closed n'gons. 



(3) If one of the points be the first extremity of a closed w'gon, and an- 

 other of them be first extremity of a closed 2w'gon. Draw the closing chord 

 of the two open w'gons composing the closed 2w'gon to pierce the tangent- 

 plane at the first extremity of the closed w'gon in the point p ; in the clo- 

 sing chord find the point /x which is conjugate to the point of puncture p^; 

 through /i and the first extremity of the closed w'gon draw the line xxy and 

 find the line ii reciprocal to xx. Then will the lines xx and ii pierce the 

 quadric in the four points which constitute the first extremities of all the 

 inscribable closed w'gons. 



(4) If two of the assumed points be found to be first extremities of closed 

 2w'gons, we may find the first extremities of the inscribable closed n'gons 

 by either of the four following methods : — 



First method. — Draw the two closing chords of the open w'gons compo- 

 sing the 2/i'gons ; and, if these chords intersect in a point p, draw the line 

 XX which is polar of p in respect to the trace of the plane of the two chords ; 

 find the line ii reciprocal to xx. Then will xx and ii pierce the quadric in 

 the first extremities of the closed Tj'gons. But if the chords do not inter- 

 sect, draw tangent-planes at their extremities ; find the two pairs of points 

 (one pair in each chord) which divide these closing chords and the seg- 

 ments intercepted by the tangent-planes harmonically ; draw the line xx 

 through the two of these jioints which divide the chords internally ; draw 

 the line ii through the two points which divide the chords externally. 

 Then will xx and ii be reciprocal lines piercing the quadric in four points 

 which are first extremities of the inscribable closed ?i'gons. 



Second method. — Find a^a^, b^b^y c^c^ the closing chords of three open 

 ^'gons. If any two of these intersect, proceed as in the last method ; but 

 if not, proceed as follows : — In the chord a^a^ find the point m which cor- 

 responds to infinity (on the same line) in one of the homographic figures 

 in which the extremities of all inscribable T^'gons are corresponding in- 

 terchangeable points ; in the same chord a^^a^ find the two points x and i 

 such that mx=:^mi= V ma^ . ma^ ; through x draw the line xx which cuts 

 the two non-planar chords b^b^, c^c^ ; through i draw the line ii which cuts 

 the same two non-planar chords b^b^, c^c^. Then will xx and ii be reci- 

 procal lines piercing the quadric in the four points which are the first extre- 

 mities of closed n'gons. 



