2 INTRODUCTION. 



If we make y l = px 1} y 2 = p# 2 , 2/s = P x a&amp;gt; 2/4 = P x * we can eliminate x l ,x z , x a , 

 \ and obtain the following biquadratic for p, 



= 0. 



The four roots of this indicate four double points, i.e. points which remain 

 unaltered. But these points are not necessarily distinct or real. 



What we have written down is of course the general homographic trans 

 formation of the points in space. For the displacement of a rigid system is 

 a homographic transformation of all its points, but it is a very special kind of 

 homographic transformation, as will be made apparent when we consider what 

 has befallen the four double points. 



In the first place, since the distance between every two points before the 

 transformation is the same as their distance after the transformation it 

 follows that every point in the plane at infinity before any finite trans 

 formation must be in the same plane afterwards. Hence the plane at infinity 

 remains in the same position. Further, a sphere before this transformation is 

 still a sphere after it. But it is well known that all spheres intersect the 

 plane at infinity in the same imaginary circle H. Hence we see not only that 

 the plane at infinity must remain unaltered by the transformation but that a 

 certain imaginary circle in that plane is also unaltered. 



A system of points P lt P 2 , P 3 , &c. on this circle fl will, therefore, have 

 as their correspondent points Q,, Q.,, Q 3 , &amp;lt;foc. also on fl As all anhar- 

 monic ratios are unaltered by a linear transformation it follows that the 

 systems P lt P 2 , P 3 , &c. and Q l} Q 2 , Q :t , &c. are homographic. There will, 

 therefore, be two double points of this homography, O l and 0, and these will 

 be the same after the transformation as they were before. They are, there 

 fore, two of the four double points of which we were in search. 



It should be remarked that the points Oj and 0., cannot coincide, for if 

 they coincide in 0, then must be the double point corresponding to a 

 repeated root of the biquadratic for p. But such a root is real. Hence 

 must be real. But every point on fl is imaginary. Hence this case is 

 impossible. 



As Q is unaltered and Oi and 2 are fixed, the tangents at 0, and 2 

 are fixed, and so is therefore T, the intersection of these tangents; this is 

 accordingly the third of the four points wanted. It lies in the plane at 

 infinity, but is a real point. The ray O^Og, is also real ; it is the vanishing 

 line of the planes perpendicular to the parallel rays, of which T is the 

 vanishing point. 



