E^rCH: PROJECTIVE GROUPS. 1 3 



k=o, k-f- 00, k= — 

 o 



(_i,'-). For k=o take for K any conic tangent to 1, and for K^ a 

 degenerated ellipse EF in 1 and tangent to K^. Obviously q^ lies 

 in 1 and C in V (Fig. 8), hence k=o. As the construction shows, 

 to each point A of the plane corresponds a point A^ of the axis. 



Conversely, to each point A', except the points of 1, corresponds 

 the centre C, i. e., the whole plane of the other system corresponds 

 to the centre C. To each ray g through C corresponds its point 

 of intersection with 1. 



{h). For k= 03 results an analogous case in which r coincides 

 with 1 and C with q^ 



(/). For k^ — , centre, axis, and counter-axis of collineation are 

 o 



coincident and the conies K and K' indeterminate. The points of 



each system correspond to the centre C. To each straight line in 



either system corresponds the axis 1 and to each ray through the 



centre in one system all the rays through the centre in the other 



system. 



(/') *The last important case which is to be considered here is 

 characterized by k = -f-i ^i^d the centre C in the finite part of the 

 axis 1. The coiuiter-axes q^ and r are opposite and equidistant 

 from 1. These conditions are realized by two conies K and K^ 

 related by involution, but of wdiich one conic is revolved about the 

 axis of collineation into the half plane of the other. After having 

 revolved one half-plane into the other, one or two common tan- 

 gents of K and K^ coincide with 1. The center C becomes accord- 

 ingly the intersection-point of the other common tangent with 1, or 

 the common point of tangency of K and K^. In the first case the 

 two conies have a contact of the second, in the other a contact of 

 the third order. If no special assumption about the position of 

 the centre and the conies is made the two conies will be in double 

 contact, if, however, the connection line of the centre and the point 

 of tangency of the conies in involution is perpendicular to the axis 

 1 the case of a triple contact arises. Figs, g and lo illustrate these 

 two cases. 



What has been found here by logical deduction from the laws of 

 collineation can also be proved by assuming one conic tangent to 

 1, the center on 1, the counter-axes q^ and r opposite and equidis- 

 tant from 1 and by constructing the corresponding conic. As the 



*This is what Sophus Lie in his "Vorlesungen ueber continuierliche Gruppeu" calls 

 elation. In accordance with Lie we put this case at the end of the classification. 



