158 Professor Baker, On the reduction of homography 



On the reduction of homography to movement in three dimensions. 

 By Professor H. F. Baker. 



[Read 9 February 1920.] 



It is known, and was assumed by Poncelet, that a self-homo- 

 graphy of a Euclidian metrical plane is reducible, by a "move- 

 ment" of one of the figures, to a perspectivity, called by Poncelet 

 a homology, that is a transformation in which the join of two 

 corresponding points passes through a fixed point, while corre- 

 sponding lines meet on a fixed line. It is also known that this is 

 not true for Euclidian metrical space of three dimensions. See 

 H. J. S. Smith, Proc. Lond. Math. Soc, ii, 1866-1869, pp. 196-248; 

 Chasles, Geom. Sup., 1880, pp. 375-381; Salmon, Higher Plane 

 Curves, 1879, p. 298. 



The question arises what is the nearest analogous proposition 

 for three dimensions. In the projective plane the reduction can 

 also be made (instead of by a movement), by means of a trans- 

 formation keeping a given arbitrary conic unaltered. In the present 

 note we prove that in projective space of three dimensions a general 

 homography is reducible, by means of a transformation leaving 

 an arbitrary quadric unaltered, apphed to one of the figures, to 

 a transformation which we may call an axis-range perspectivity; 

 namely one in which every point of a certain range is unaltered 

 and every plane passing through a certain axis is unaltered. This 

 seems to add something to what is known, and to be a very natural 

 generahsation of the results in a plane. 



§ 1. Let S, Sq be two quadrics with a common seK-polar tetra- 

 hedron ; let the two systems of generators of Sq be called the axes 

 and transversals, respectively. There are four axes of Sq which 

 touch the curve of intersection of S and Sq, and are generators of 

 the developable surface of common tangent planes of S and Sq] 

 there are also four transversals of Sq with the same property. The 

 eight points of contact of these lines with the curve {S, Sq) are on 

 the curve upon S where it is touched by the common tangent 

 planes of S and Sq] they are then the common points of S, Sq, Sq', 

 where Sq is the polar reciprocal of Sq in regard to S. 



§ 2. We now consider a certain porismatic relation between the 

 two quadrics. From an axis of Sq, which we distinguish by the 

 parameter 6, two tangent planes can be drawn to S, each of which 

 will contain a transversal of Sq, say these are distinguished by the 



