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PROBLEMS AND SOLUTIONS. 
595 
equations which put in evidence the double contact with the three critical conics 
respectively. We have also, identically, 
A = (x + y + z + w) (x + y — Sz — w) — 2w (x + y — z — w) + 4 (z- — xy), 
and the equation A = 0 may therefore be written 
A = — 2w (x + y — z — w) + 4 (z- — xy) = 0, 
a form which shows that the conic z 2 — xy = 0 meets the line w — 0 in the same two 
points in which it is met by the conic A = 0. And it hence appears by symmetry that 
the conics 
A = 0, x 1 — yz — 0, y- — zx = 0, z 2 — xy = 0 meet the line w = 0 in the same two points, 
A = 0, w 2 — yz = 0, y- — zw = 0, z 2 — wy = 0 meet the line x = 0 in the same two points, 
A = 0, w- — xz = 0, x- — zw = 0, z 2 — wx — 0 meet the line y = 0 in the same two points, 
A = 0, w- — xy = 0, x- — yw = 0, y- — ivx = 0 meet the line z = 0 in the same two points, 
which are the relations constituting the latter part of the proposed theorem. 
2. The analogous theorems in space may be briefly referred to. Taking x — 0, 
y = 0, z = 0, w = 0 as the equations of the faces of a tetrahedron A BCD, then the 
implicit constants may be so determined that the coordinates of a given arbitrary point 
0 shall be (1, 1, 1, 1). We may by lines drawn from the vertices of the tetrahedron 
project the point 0 on the faces, so as to obtain a point in each of the four faces; 
and then in each face, by lines drawn from the vertices of the face, project the point 
in that face upon the edges of the face; the two points thus obtained on each edge 
of the tetrahedron are (it is easy to see) one and the same point; that is, we have 
on each edge of the tetrahedron a point; and there exists a quadric surface 
A = x- + y- + z* + w- — Zyz — 2 zx — 2 xy — 2 xw — %yw — 2zw = 0 
touching the edges of the tetrahedron in these six points respectively. 
The surface in question has plane contact with 
the hyperboloid xy + zw = 0 along the intersection with x + y—z — w = 0, 
„ „ xz+yw= 0 „ „ „ x-y + z-w = 0, 
„ „ xw + yz = 0 „ „ „ x — y — z + w = 0, 
and moreover the surfaces 
A = 0, x 2 — yz = 0, y l — zx = 0, z" — xy —0 meet the line w — 0, x + y + z + w = 0 
in the same two points; 
A = 0, w 2 — yz = 0, y 2 —zw = 0, z 2 — wy — 0 meet the line « = 0, x + y + z + w = 0 
in the same two points; 
A — 0, w 2 — xz = 0, x 2 — zw = 0, z 2 — wx — 0 meet the line y = 0, x + y + z + w = 0 
in the same two points ; 
A = 0, w 2 — xy — 0, x 2 — yw — 0, y 2 — wx = 0 meet the line z = 0, x + y + z + w — 0 
in the same two points. 
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