BY MARTIN GARDINER, C.E. 123 



Analysis of a second method of solution. 



From theorems 9 and 16 we learn that according as we can 

 inscribe one open 2 w'gon or one closed 2 w'gon, so will the 

 problem of the inscription of the closed w'gons be non-porismatic 

 or partially porismatic. 



When the data is in the non-porismatic state, it is evident 

 that if we inscribe an open 2 n'gon, and draw the plane which 

 contains its extremities and the first point of its n -f 1 th side ; and 

 that we then inscribe another open 2 w'gon whose first extremity 

 is not in this plane ; then will the plane through the extremities 

 and first point of the n + 1 th side of this last 2 w'gon cut the 

 other plane in a straight line xx which pierces the surface in the 

 points (real or imaginary as may be) which are the answerable 

 positions for the first extremities of the inscribable closed 

 n'gons. 



When we can inscribe a closed 2 w'gon ; it is evident that we 

 can inscribe open w'gons, and that the closing chords of these will 

 intersect in a point the trace of whose polar plane is the locus of 

 the answerable positions for the first extremities of the closed 

 w'gons. 



When the problem is fully porismatic, the fact will be intima- 

 ted to us by our being enabled to inscribe 4 closed w'gons whose 

 first extremities are not all in one plane. 



^g^ This method of solution is also complete, and is applica- 

 ble to the following more general problem : " Given a surface S 

 of the second degree, and n entities in prescribed order, each entity 

 being either a given point, or a conicoid having double contact with S; 

 to inscribe in the surface S closed n'ffons such that each side of each 

 rigon ic ill meet with the entity of the series ivhich is of like rank in 

 the series with such side in the rigon, and in such a manner as to 

 pass through the entity if it be a point, or to be tangent of certain 

 prescribable rotative to the trace made on the conicoid by a plane 

 containing the Jirst point of such side and the line of contact of the 

 conicoid ivith S if it be a conicoid" 



Third Method of Solution. 

 The following method of finding the first extremities of the 



