emch: projective groups. 



nating the points of intersection of an}' ray of the congruence with 

 the two osculating planes as corresponding points, the two planes 

 are coUinear and we have immediately our general theorem if we 

 revolve one of 'the osculating planes into the other about their 

 line of intersection. In the case of a perspective collineation 

 the congruence is of the order i and the class o. The develop- 

 able surface is not determinate, so that we may choose a cone 

 of the 2. class which with any two planes determines a perspective 

 collineation. 



As in the general case of projectivit}', to each point P the cor- 

 responding Pi is obtained b}' drawing two tangents from P to the 

 conic K which will intersect the line 1 in two points; from these 

 two points draw the tangents to the conic K^. Their point of 

 intersection gives the required point Pi. 



That this construction gives perspective collineation we can also 

 prove without referring to the general case of projectivity. 



Assume the two conies K and K^ in the required position 

 (Fig. i) and draw the common tangents t^ and t^ which intersect 

 each other in C. 



K and K^ now belong to a system of conies tangent to t^ and t^ 

 and to 1 at a fixed point T. The polars p, p^ ... .of C in regard 

 to the conies K, Ki,....of the system, therefore, intersect each 

 other in one and the same point S on 1, and if we draw any other 

 line t through C which intersects K in A and B, and K^ in A^ and 



