46$ Prof. Cayley on a Theorem relating 



or, what is the same thing, we have with the points 1, 2, 3, 4 

 and the point 0' constructed the four points P, Q, B, S such that 



IS, 2B, 3Q, 4P meet in a, 



2S, IE, 4Q, 3/3 „ /3, 



3S, 4E, 1Q, 2P „ y. 

 The eight points 1, 2, 3, 4, P, Q, E, S form a figure such as the 

 perspective representation of 

 a parallelopiped, or, if we 

 please, a cube ; and not only 

 so, but the plane figure is 

 really a certain perspective 

 representation of the cube ; 

 this identification depends 

 on the following two theo- 

 rems : — 



1. Considering the four 

 summits 1, 2, 3, 4 , which 

 are such that no tw r o of them 

 belong to the same 'edge, 

 then, if through any point O 

 we draw 



the line OA' meeting the lines 41, 23, 



„ OB' „ „ 42, 31, 



„ OC „ „ 43, 12, 



and the lines Oa, 0/3, O7 parallel to the three edges of the cube 

 respectively, the three planes (OA', Oa), (OB' 0/3), (OC, O7) 

 will meet in a line. 



2. For a properly-selected position of the point 0, 



the lines OB', OC, Oa will lie in a plane, 

 „ OC, OA', 0/3 „ „ 

 „ OA', OB', 7 „ „ 



In fact for such a position of O, projecting the whole figure on 

 any plane whatever, the lines 01, 02, 03, 04, OP, OQ, OE, 

 OS, Oa, 0/3, O7, OA', OB', OC meet the plane of projection in 

 the points 1, 2, 3, 4, P, Q, E, S, a, /3, 7, A', B', C related to 

 each other as in the last-mentioned form of the plane theorem. 

 To prove the two solid theorems, take for the origin, Oa, 0/3, 

 O7 for the axes, (a, /3, y) for the coordinates of the summit S, 

 and 1 for the edge of the cube, 



the coordinates of 1 are a+ 1, /3, 7, 



» 2 „ *, &+'}, y, 



„ 3 „ a, ft 7 + I, 



4 „ a + 1, £+!, 7 + I. 



