40 Analytical Solution. 



These being evidently linear functions of the co-ordinates of 

 k, we know that the first and final extremities of any inscribed 



n'gon are correspondents in a pair of hoinographic figures ^ , ^ . 



1111 

 Denote these co-ordinates by X, Y, Z, W. 



For all closed n'gons snbject to the imposed conditions the 



first and final extremities are coincident, and therefore for such 



n'gons we must have 



1 i 1 1 

 a?:y:£:t0::X:Y:Z:~W; 



And from these we at once obtain the equations : 



(S...S)(a y— /3 tf)A + (S...S)0 y— j8 ar) A+...+(S)(a y— /S #)A 



n 2 1 11 n 3 2 ' 2 2 n n-1 n-1 n-1 



+ (a y—(3 X) A = 



($...8)(az—yx) A+(S...S)(«2- 7 ir) A+... + (S)(a z— y a?) A 



n 2 1 1 1 n 3 2 2 2 n n-1 n-1 n-1 



+ (a Z — y X) A = 



(S...S)(ffiw-S.i:)A + (S...S)(««o- 8«)A+... + (S)(aitf- S a;)A 



n 2 1 1 1 n 3 2 2 2 n n-1 n-1 n-1 



+ «0— 8 a) A = 

 n n n 



These are evidently the equations of three hyperboloids of one 

 sheet, whose common intersection contains all the double points 

 of the homographic figures which are first extremities of the 

 inscribable closed n'gons. 



Again, if we proceed from the point k with another n'gon in- 



i 

 scribed in a reverse manner through the given points, we can in 



like manner arrive at analagous expressions for the- co-ordinates 



of the final extremity as linear functions of the co-ordinates of 



the first extremity k ; and indicating the co-ordinates of the final 



i 

 extremity by X, T, Z, "W, it is obvious that for all double and 



1111 

 interchangeable points in the homographic figures, we must have 



the relations 



X:Y:Z:W::X:Y:Z:W 



llli 



from which we can obtain equations of three hyperboloids of one 



