 By Elisabeth Hébert (ed.)

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Then the normals are parallel and the three planes are parallel to one another. However, if R(V) = 2 the three points at infinity just mentioned are distinct and so the three planes intersect in a finite line. Case III. Let R(U) = 1. We verify that P2ui — /> A K Pi K P2 x+ μι y+ pi 1*2 P2 »i 0. Pi V2 P2 That is, ux = 0 and u2 = 0 are the same plane. Similarly u3 = 0 also gives the same plane. The three planes are accordingly coincident. These results are conveniently summarized in the following table : R A N K OP MATBIX U 3 tf 3 Planes have a unique finite point of intersection 2 Planes parallel to a line (planes form a prism) 1 2 Pencil of planes with a finite axis Parallel planes 1 >H Coincident planes E x a m p l e 3 .

2 . Find the condition t h a t the two circles x2 -\- y2 -{- z2 -\- 2pxx + 2qxy -f 2rxz + dx = λχχ + μ\$ + νλζ + Pl = 0 and x2 + 2/2 + z2 + 2p 2 # + 2g2y + 2r2z + d2 = λ2χ + μ22/ + ^2Z + p2 — 0 should lie on the same sphere. 3 . PQ is the shortest distance between the two straight lines AP and BQ. If A B is of constant length as A and B vary, show t h a t the radius of the sphere ABPQ is constant. 4 . )/m = (z — y)\n should touch the sphere x2 + y2 -+- z2 + 2p# + 2gy + 2rz -f d = 0. Hence find the locus of the centres of spheres of constant radius which pass through a given point and touch a given straight line.

These two equations simplify to x2 + y2 + z2 = w = 0. 2). Since all plane sections of a sphere are circles, we call this curve the circle at infinity or absolute and it is generally denoted by Ω. 2). Hence/ — g = h = 0 and a = b = c. Thus the surface is a sphere. The straight line at infinity in a plane cuts any circle of that plane in the two circular points of the plane. A plane cuts the circle at infinity in the two points of intersection of the line at infinity in the plane and Ω. Thus a plane cuts Ω in the two circular points lying in the plane.