392 



Mr. M. Gardiner on Undevelopable [May 14, 



tiesofinscribable closed 2«'gons (real or imaginary according as the quadric 

 S is ruled or convex). 



(4) If one of the three chosen points be the first extremity of an open 

 2?i'gon (no matter as to the other two points) the problem is non-porismatic, 

 and we can find the first extremities of the closed «'gons by either of the 

 four following methods : — 



First method. — Continue the 2w'gon until a 4w*gon be formed, and draw 

 the plane P which contains the extremities of this 4w'gon and the point of 

 junction of the two open 2w'gons composing it. Assume another point in 

 the surface, not in the trace of the plane P, and, making it a first extremity, 

 inscribe another 4w'gon ; and through the extremities of this 4w'gon and 

 the point of junction of the two open 2w'gons composing it draw a plane Q. 

 Then with the line xx of intersection of the planes P and Q pierce the 

 quadric in the only points (real or imaginary as may be) which are first 

 extremities of closed w'gons ; and the line ii reciprocal to xx will pierce 

 the quadric in points (real or imaginary as may be) which are first extre- 

 mities of closed 27i'gons. 



Second method. — By the additional inscription of another open Ti'gon 

 convert the open 2«'gon into an open 3»'gon, and put A, B, C to represent 

 the three open w'gons composing the open 3«'gon. Find the point of punc- 

 ture of the hne through the first extremity of A and the final extremity of 

 B with the tangent-plane at the junction of A and B ; find the point of 

 puncture of the line through the first extremity of B and the final extre- 

 mity of C with the tangent-plane at the junction of B and C. Then will 

 the line xx through the two points of puncture pierce the quadric in two 

 points (real or imaginary as may be) which are the first extremities of the 

 only inscribable closed w'gons ; and the line ii, which is reciprocal to xx, 

 will pierce in first extremities of closed 2?i'gons. 



N.B. When S is a hyperboloid of one sheet, and that the first extremi- 

 ties of the closed ?z'gons are real, then the first extremities of the closed 

 2?i'gons are also real ; and it is evident there are two pairs of generators 

 which are corresponding interchangeable lines in the homographic figures 

 in which the extremities of the inscribable w'gons are pairs of corresponding 

 points. It is moreover evident that when all the corresponding points of 

 such figures are not interchangeable, these are the only pairs of inter- 

 changeable generators ; and we must not assume the first extremities of the 

 4/2'gons or 3?z'gons in these lines. 



Third method. — Let o^, o^, o^, . . . On be the n fixed points through which 

 the sides must pass in order. 



Assume the constant homological ratio — 1 for homological systems, and 

 making o^ vertex and its polar plane axis, find the point homological to 

 the centre of the quadric ; assume o^ as vertex and its polar plane as 

 axis, and find the point cc^ homological to ; and proceed thus directly in 

 order through the w points until arrived at the point Assume o„ as 

 vertex and its polar plane as axis, and find the point a_ i homological to the 



