INTRODUCTION. 3 



In general in any homographic transformation there cannot be four distinct 

 double points in a plane, unless every point of the plane is a double point. 

 For suppose P 1; P 2 , P 3 , P 4 were four distinct coplanar double points and that 

 any other point R had a correspondent R 1 . Draw the conic through P lt P 2 , 

 P 3 , P 4 , R. Then R must lie on this conic because the anharmonic ratios 

 R(P lt P 2 , P 3 , P 4 ) and R (l\, P 2 , P 3 , P 4 ) are equal. We have also 

 P,(P 2 , P 3 , P 4) R) and P, (P 2 , P 3 , P 4 , R ) equal, but this is impossible if R 

 and R be distinct. R is therefore a double point. 



In the case of the displaced rigid body suppose there is a fourth distinct 

 double point in the plane at infinity. Each ray connected with the body 

 will then have one double point at infinity, so that after the transformation 

 the ray must again pass through the same point, i.e. the transformed position 

 of each ray must be parallel to its original position. This is a special form of 

 displacement. It is merely a translation of the whole rigid system in which 

 every ray moves parallel to itself. 



In the more general type of displacement there can therefore be no 

 double point distinct from T, 0], 2 and lying in the plane at infinity. Nor 

 can there be in general another double point at a finite position T . For if 

 so, then the ray TT is unaltered in position, and any finite point T&quot; on the 

 ray TT will be also unaltered, since this homographic transformation does 

 not alter distances. Hence every point on TT is a double point. Here 

 again we must have fallen on a special case where the double points instead 

 of being only four have become infinitely numerous. In this case every point 

 on a particular ray has become a double point. The change of the body from 

 one position to the other could therefore be effected by simple rotation around 

 this ray. 



There must however be four double points even in the most general case. 

 Not one of these is to be finite, and in the plane at infinity not more than 

 three are to be distinct. The fourth double point must be in the plane at 

 infinity, and there it must coincide with either 0,, 2 or T. Thus we learn 

 that the most general displacement of a rigid system is a homographic trans 

 formation of all its points with the condition that two of its double points are 

 on the imaginary circle fi in the plane at infinity, while the pole of their chord 

 gives a third. Of these three one, we shall presently see which one, is to be 

 regarded as formed of two coincident double points. 



All rays through T are parallel rays, and hence we learn that in the 

 general displacement of a rigid body there is one real parallel system of 

 rays each of which L is transformed into a parallel ray L . Let A be any 

 plane perpendicular to this parallel system. Let L and L cut A in the points 

 R and R. Then as L and L move, R and R are corresponding points in two 

 plane homographic systems. Any two such systems in a plane will of course 



1-2 



