a Surface of the Second Ordei- into itself. 209 



it is touched by the two cones, 6, (f) ; the cones themselves may, 

 it is clear, be spoken of as the cones 6, (f>. And let the two 

 points be P, Q. The line PQ touches the two cones, it is there- 

 fore the line of intersection of the tangent plane through P to 

 the cone 6, and the tangent plane through P to the cone 0. Let 

 one of the generating lines through P meet the section 6 in the 

 point A, and the other of the generating lines through P meet 

 the section </> in the point B. The tangent planes through P to 

 the cones 6, respectively are nothing else than the tangent 

 planes to the surface U at the points A, B respectively. We 

 have therefore at these points two generating lines meeting in 

 the point P ; the other two generating lines at the points A, B 

 meet in like manner in the point Q. Or P, Q are opposite 

 angles of a skew quadrangle formed by four generating lines 

 (or, what is the same thing, lying upon the surface of the second 

 order), and having its other two angles, one of them on the sec- 

 tion 6 and the other on the section </>; and if we consider the 

 side PA as belonging determinately to one or the other of the 

 two systems of generating lines, then when P is given, the corre- 

 sponding point Q is, it is clear, completely determined. What 

 precedes may be recapitulated in the statement, that in the 

 improper transformation of a surface of the second order into 

 itself, we have, as corresponding points, the opposite angles of a 

 skew quadrangle lying upon the surface, and having the other 

 two opposite angles upon given plane sections of the surface. I 

 may add, that attending only to the sections through the points 

 of intersection of 6, (j), if the point P be situate anywhere in one 

 of these sections, the point Q will be always situate in another 

 of these sections, i. e, the sections correspond to each other in 

 pairs ; in particular, the sections 9, are corresponding sections, 

 so also are the sections 0, $ (each of them two generating lines) 

 made by tangent planes of the surface. Any three pairs of sec- 

 tions form an involution ; the two sections which are the sibi- 

 conj agates of the involution are of course such, that, if the point 

 P be situate in either of these sections, the cori'esponding point 

 Q will be situate in the same section. It may be noticed that 

 when the two sections 6, <^ coincide, the line joining the cor- 

 responding points passes through a fixed point, viz. the pole of 

 the plane of the coincident sections ; in fact the lines PQ and 

 AB are in every case reciprocal polars, and in the prsent case the 

 line AB lies in a fixed plane, viz. the plane of the coincident 

 sections, the line PQ passes therefore thi'ough the pole of this 

 plane. This agrees with the remarks made in the first part of 

 the present paper. 



The analytical investigation in the case where the surface of 

 the second order is represented under the form xy—zw=:0 is so 



