1868.] 



Uniquadric Homographics. 



391 



relation between either system and the perspective of the other to any 

 point on that Hne of intersection. 



Chapter V. treats of systems having their four double planes all tan- 

 gential to the quadric on which they lie, shows that the chords connecting 

 their several pairs of corresponding points touch two other quadrics having 

 quadruple contact with the original at the four double points of the systems, 

 and gives various constructions for the determination of the four double 

 points when the law connecting the several pairs of corresponding points 

 of the systems is given or known. 



Chapter VI. gives various criteria for determining in certain cases ta 

 which of the preceding classes two homographic systems belong, where, as 

 in the problem whose solution forms the principal object of the memoir, 

 the law connecting the same pairs of corresponding points of the systems 

 is given or known. 



Chapter VII. contains numerous theorems, several of much interest and 

 originality, respecting open and closed polygons inscribed in undevelopable 

 quadrics, whose sides pass in the same order of sequence through a common 

 system of points in space, all deduced from the principles established in 

 the preceding chapters, and several having direct reference to the interest- 

 ing problem to be considered in the next and closing chapter. 



Chapter VIII. — Given an undevelopable quadric and n fixed points in 

 to find the space ; first extremities of inscribable closed ?z'gons, or the locus 

 of the first extremities when the inscription of the closed w'gons is poris- 

 matic. 



When the number n of given points is odd. 

 Assume any three points a^, b^^ in the surface, no two of which are on 

 one generator, as first extremities, and proceed to inscribe 2^i'gons. 



(1) If the three points be found to be first extremities of closed w*gons, 

 then will the trace of their plane be the locus of first extremities of closed 

 w'gons, the problem in such case being partially porismatic. 



(2) If the points are first extremities of closed 27i'gons, or if two of them 

 be first extremities of closed 2/i'gons and the third one a first extremity of 

 a closed w'gon, or if one of the points be the first extremity of a closed 

 2/i'gon, and the other two points first extremities of closed w*gons, then 

 the line or lines forming the closing chords of the open /i'gons composing 

 the 2w'gon or 2/i'gons (as may be) and the tangent plane or planes at the 

 first extremity or extremities of the closed w'gon or w'gons (as may be) 

 meet in one point p, the trace of whose polar plane R is the locus of first 

 extremities of inscribable closed ?i'gons, the problem in such case being par- 

 tially porismatic. 



(3) If two of the points be first extremities of closed w'gons and the 

 third point the first extremity of an open 2n'gon, then the problem is non- 

 porismatic, and the two closed n'gons are the only inscribable closed /I'gons. 

 Moreover the reciprocal of the line joining the first extremities of the two 

 closed rt'gons will pierce the quadric in points which are the first extremi- 



