382 



in one common point, namely, in tiie pole of the given 

 plane). 



If we seek to inscribe a polygon of 4/w sides in a surface of 

 the second order, under the condition that its opposite sides 

 shall intersect respectively in 2m given points, the quaternion 

 analysis conducts generally to two polar right lines, as loci of 

 the first corner, which lines are the same with those that would 

 be otherwise found as loci of the first corner of an inscribed 

 polygon of 2m sides, passing respectively through the 2m 

 given points. Thus, in general, the polygon of 4m sides, 

 found as above, is merely the polygon of 2m sides, with each 

 side twice traversed by the motion of a point along its peri- 

 meter. But if a certain condition be satisfied, by a certain 

 arrangement of the 2m given points in space; namely, if the 

 last point ao„, be on that real right line which is the locus of 

 the first corner of a real or imaginary inscribed polygon of 

 2m - 1 sides, which pass respectively through the first 2m - 1 

 given points Aj, . . . Asn,.,; then the inscribed polygon of 4w 

 distinct sides becomes not only possible but indeterminate, 

 its first corner being in this case allowed to take any position 

 on the surface. For example, if the two triangles p' p', p'a, 

 p" p'\ p'a be inscribed in a conic, so that the corresponding 

 sides p' p'l and p" p"i intersect each other in a,; p', p'2 and 

 p"i v'i in A3 ; and p'2 p', p'a p", in A3 ; and if we take a 

 fourth point A4 on the right line p' p", and conceive any sur- 

 face of the second order constructed so as to contain the given 

 conic; then any point p, on this surface, is fit to be the first 

 corner of a plane or gauche octagon, p p, . . . P7, inscribed in 

 the surface, so that the first and fifth sides P p^ P4 P5 shall 

 intersect in Aj ; the second and sixth sides in ao; the third and 

 seventh sides in A3 ; and the fourth and eighth in A4. And 

 generally if 2m given points be points of intersection of oppo- 

 site sides of any one inscribed polygon of Am sides, the same 

 2m points are then fit to be intersections of opposite sides of 

 infinitely many other inscribed polygons, plane or gauche, of 



