The Philosophical magazine; a journal of theoretical, experimental and applied physics . e the co-ordinates of the point 0, then the coordinates of (a, /3, 7) arefound to be (0, Y, Z), (X, 0, Y), (X, Y, 0) respectively. Hence the equations of the lines Aa, B/S, C7 arerespectively X, y , z =, Y, Z0, Y, Z that is, x{ YZ-YZ)+y(a;(-YZ )+y{ X, X, -Y, 0, = 0, X ,y ,X,Y,X, Y, = 0; ZX)+„-(-XY ) = 0, ZX-ZX) + .-( XY)=0, a:{ +YZ)+2/(-ZX )+.-( XY-XY) = 0, which are obviously the equations of three lines which meet in apoint. But the theorem may be exhibited as a theorem relating to aquadrangle 1234 a
The Philosophical magazine; a journal of theoretical, experimental and applied physics . e the co-ordinates of the point 0, then the coordinates of (a, /3, 7) arefound to be (0, Y, Z), (X, 0, Y), (X, Y, 0) respectively. Hence the equations of the lines Aa, B/S, C7 arerespectively X, y , z =, Y, Z0, Y, Z that is, x{ YZ-YZ)+y(a;(-YZ )+y{ X, X, -Y, 0, = 0, X ,y ,X,Y,X, Y, = 0; ZX)+„-(-XY ) = 0, ZX-ZX) + .-( XY)=0, a:{ +YZ)+2/(-ZX )+.-( XY-XY) = 0, which are obviously the equations of three lines which meet in apoint. But the theorem may be exhibited as a theorem relating to aquadrangle 1234 and a point 0; for writing 1, 2, 3, 4 inplace of A, B, C, 0, the triangle ABC is in fact the triangleformed by the three centres , , of the qua-drangle 1234, hence the triangle in question must be similarlyrelated to each of the four triangles 423, 431, 412, 123; or,forming the diagram Q3 4 1 3, we have the following form of the theorem: viz. the lines a4, /33, 72 meet in a point P, «3, y84, 7I „ „ Q, «2, /31, 74 „ „ B, al, ^2, 73 „ „ S,. 463- Prof. Cayley on a Theorem relating or, what is the same thing, we have with the points 1, 2, 3, 4and the point O constructed the four points P, Q, R, S such that IS, 2R, 3Q, 4P meet in u, 2S, IR, 4Q, 3^ „ /9, 3S, 4R, IQ, 2P „ 7. The eight points 1, 2, 3, 4, P, Q, R, S form a figure such as theperspective representation ofa parallelopiped, or, if weplease, a cube; and not onlyso, but the plane figure isreally a certain perspectiverepresentation of the cube ;this identification dependson the following two theo-rems :— 1. Considering the foursummits 1, 2, 3, 4, whichare such that no two of thembelong to the same edge,then, if through any point Owe 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 cuberespectively, the three planes (OA, Oa), (OB 0/3), (OC, O7)will meet in a line. 2. For a pro
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