Analytical Solution. 39 



points taken in order : and putting A, A, A for the functions 



12 n 



indicating the polar planes of the n given points when the varia- 

 ble co-ordinates are replaced therein by the co-ordinates of k, k, 



1 2 



k respectively, we can easily express the co-ordinates of the 



n 



points k, k, k as linear functions of those of the first 



2 3 n + l 



extremity k of the n'gon. For let x, y, z, to, be the co-ordinates 

 of the point k, and it is evident the co-ordinates of k are propor- 



1 2 



tional to 



S x — 2 A a; 

 ill 



S y — 2 A ; 



i i i 



S Z — 2 A y ; 



I 11 



S w — 2 A 8. 

 i i i 



And from these we have, in like manner, the co-ordinates of 

 the point k proportional to 



3 



(S S) x — 2 (S a) A — 2 (a) A ■ 



2 1 2 11 2 2 



(SS) y — 2(S /3) A— 2 (0) A ; 



2 1 ' 2 11 2 2 



(S S) z — 2 (S y) A — 2 (y) A ; 

 2 1 2 11 2 2 



(S S) t6> — 2 (S 8) A — 2 (8) A. 



2 1 2 11 2 2 



And proceeding in like manner with the remaining sides of the 

 n'gon, we find the co-ordinates of k proportional to 



n + 1 



(S...S) as— 2(8... S a) A— 2 (S...S «)A— ...— 2(S a) A — 2(a) A; 



nl n211 n322 nn-ln-1 nn 



(S...S)y— 2(S...S/3)A— 2 (S...S/3)A— ...— 2(8 0) A — 2(0) A; 



n 1 n 2 1 1 n 3 2 2 n ii-l n-1 n n 



(S...S) z— 2(S...S y)A— 2 (S...S.y)A— ...— 2(8 y) A — 2(y)A; 

 nl n211 n322 n n-1 n-1 n n 



(S...S) w— 2(8... 8 S)A— 2 (S...S S) A— ...— 2(8 8) A —2 (8) A. 



n 1 n 2 1 1 n 3 2 2 nn-ln-1 n n 



