to movement m three dimensions 161 



containing one of each, touches the quadric S if 



pd^ + jfp'gu -{- p {fu + gv — h^) + fv = 0; 



this is the form of the (2, 2) relation in this case, the two axes 9, <f> 

 of Sq being 6, — 6. For the axis which is the line zt we have 6 = co 

 and the two corresponding values of p, q are p = 0, q = oo . It is 

 then material to remark that the transformation , 



_h - Oq h + Bq _ _ 



changes the particular axis of Sq determined by 6q into the axis 

 determined by — Oq, and changes the plane {Oq, p), or 



{h - 9q) X +fz + vt + p[{h + Bq) y + gz + ut]^ 0, 



into the plane (— Oq, p), for all values of p. This transformation is 

 one which leaves the quadric S, or xy — zt = 0, unaltered, and 

 leaves every point of the line zt, which is a generator of Sq, un- 

 altered, as well as every plane through the line xy. It is thus a 

 transformation of the kind which we have called an axis-range 

 perspective, or, in a usual phraseology, in regard to S as the 

 absolute quadric, it is what is called a "rotation" round the line zt. 



Thus we may say : // a quadric S be tetrahedrally inscribed in a 

 quadric Sq, it is possible to find a self-transformation of S, which is 

 a rotation about a generator of Sq, such^ as to change any axis of Sq 

 into its associated axis, and to change any plane through this into the 

 plane through the associated axis containing the same transversal of Sq 

 as the former. 



It is easy to write down a corresponding result for the plane. 



§ 3. Now consider any homography in space, say 

 x'= a^x + b^y + qe + d^t, ..., t'= a^x + b^y + c^z + d^f, 

 which we denote, in matrix notation, by 



{x', if, z', t') = «!, 6i, Ci, d^ {x, y, z, t). 



or also by {x') = ^ (x). Take an arbitrary quadric S; this is changed 

 by the transformation into another quadric, and is itself obtainable 

 from another by the transformation ; regarded in these two aspects 

 let it be denoted respectively by a and p', the quadric into which 

 it is changed by the transformation being a', and that from which 

 it may be supposed to arise being p. Denote the developable 

 surface formed by the common tangent planes of p and a by 

 {p, a), and that formed by the common tangent planes of p' and a 



VOL. XX. PART I. 11 



