381 



octagon, &c., when four, six, eight, &c. points are given on 

 its successive sides, two real right lines are found, which (as 

 above stated) are polars of each other ; and therefore, if the 

 surface be an ellipsoid, or a hyperboloid of tiuo sheets, the 

 problem admits generally of two real and of two imaginary 

 solutions : while if the surface be a hyperboloid of one sheet, 

 the four solutions are then, in general, together real, or toge- 

 ther imaginary. 



When a gauche pentagon, or polygon with 2m + 1 sides, 

 is to be inscribed in an ellipsoid or in a double-sheeted hyper- 

 boloid, and when the single straight line, found as above, lies 

 wholly outside the surface, so as to give two imaginary solu- 

 tions of the problem as at first proposed, this line is still not 

 useless geometrically ; for its reciprocal polar intersects the 

 surface in two real points, of which each is the first corner of 

 an inscribed decagon, or polygon with 4??? + 2 sides, whose 

 2m + 1 pairs of opposite sides intersect each other respectively 

 in the 2m + I given points, a,, Ao, ... Ao„,,,. Thus when, in 

 the well known problem of inscribing a triangle in a plane 

 conic, whose sides shall pass through three given points, the 

 known rectilinear locus of the first corner is found to have no 

 real intersection with the conic, so that the problem, as usually 

 viewed, admits of no real solution, and that the inscription 

 of the triangle becomes geometrically impossible ; we have 

 only to conceive an ellipsoid, or a double-sheeted hyperboloid, 

 to be so constructed as to contain the given conic upon its 

 surfiice; and then to take, with respect to this surface, the 

 polar of this known right line, in order to obtain two real or 

 geometrically possible solutions of another problem, not less 

 interesting : since this rectilinear polar will cut the surface in 

 two real points, of which each is the first corner of an inscribed 

 gauche hexagon whose opposite sides i?iiersect each other in the 

 three points proposed. (It may be noticed that the three 

 diajjonals of this gauche hexagon, or the three right lines 

 joining each corner to the opposite one, intersect each other 



