124 GEOMETRICAL RESEARCHES, 



inscribable closed %'gons is applicable to the more general pro- 

 blem when the data is in the non-porismatic state : 



Inscribe three w'gons such that the first extremity of the 

 second ^'gon coincides with the final extremity of the first w'gon, 

 and that the first extremity of the third n'gon coincides with the 

 final extremity of the second w'gon ; draw the plane which con- 

 tains the extremities of these ^'gons, and find its trace on the 

 given surface ; find i the point of intersection of the straight 

 line through the first extremity of the first %'gon and the final 

 extremity of the second %'gon with the tangent line to the trace 

 at the junction of these ^'gons ; find k the point of intersection 

 of the straight line through the first extremity of the second 

 ??.'gon and final extremity of the third w'gon with the tangent 

 line to the trace at the junction of these w'gons : then will the 

 points in which the straight line ik pierces the surface be the 

 answerable positions for the first extremities of the closed w'gons. 

 The proof is obyious from theorem 8, and the properties of the 

 homographic figures in which the extremities of the w'goiis are 

 corresponding points. 



21. Various simple solutions can be given to the problem 

 when all the entities are points and that the surface is either 

 spherical, cylindrical or conical. However their exhibition re- 

 quires much more room than can be accorded in this paper, so 

 that I will finish by showing how theorem 5 can be arrived at 

 when the surface is spherical. 



22. When the surface is spherical and the entities o v o^....o n 

 all points, we can easily derive theorem 5 independently of 

 homologic or homographic considerations. 



Thus. Let d t ^^ n+1 be any variable inscribed w'gon ; 

 and let ^ 2 . . . a n+v ^ ft a . . . . # n+1 , ^ a .... c w+ i> b thl *ee rc'gons 

 (inscribed at random). 



From similar triangles we immediately deduce the following 

 relations : 



