2 KANSAS UNIVERSITY QUARTERLY. 



corresponding points pass through one and the same point, the 

 projectivity, thus produced, is perspective collineation. 



In the general case to the line 1, the line of intersection of the 

 planes TT and TT^, belonging to the plane tt, corresponds a line p' 

 in the plane tt^ and to the line 1, belonging to the plane 7r\ 

 corresponds a line p in the plane tt. Connecting the corresponding 

 points of 1 and p^, and of 1 and p two conies K^ and K, in tt^ and 

 TT respectively, are produced which determine the projective trans- 

 formation. In the case of a perspective collineation, however, 

 these two conies are indeterminate; since the lines p and p^ coin- 

 cide Avith 1. We can, therefore, choose any four points A, B, C, D, 

 on 1, in TT, and connect them with their corresponding points A', 

 BV, C^, D\ iuTT^^, which coincide with the former, i. e., we can 

 draw any four lines a^, b^, c^, d^, in tt, through A, B, C, D, re- 

 spectively, which with 1 determine the conic K'. The conic K is 

 determined by those lines a, b, c, d, in tt, which correspond to the 

 lines a^, b^, c^, d^, according to the original conditions. 



Thus, the two conies K and K^ are collinear and fully deter- 

 mine the collineation. Since K^ touches the line 1, K touches 1 at 

 the same point. As it will be seen from this, the two conies K and 

 K^ characterizing the general projective transformation exist also 

 in perspective collineation; but there is a multiplicity of two conies 

 tangent to each other and tangent to the line 1 at the same point. 

 As there are (»* conies tangent to 1 one and the same perspective 

 collineation can alwa3's be represented by co^^ combinations of such 

 two conies. The line 1 is the axis of collineation, and the centre 

 C of collineation is obtained by the intersection-point of the two 

 other common tangents of the conies K and K*. 



We are now ready to make the following statement: 



Theorem i. EaeJi fiuo eonics tangent to each other determine a 

 perspective collineation with the common tangent at their point of 

 tangeficy as the axis and the intersectio?i-point of their two other 

 common tangents as the centre of collineation. 



The general theorem concerning the construction of collineation 

 by means of two conies tangent to the same line, as well as this 

 special theorem, are obtained in a natural way by studying the 

 congruence of right lines (3. i) formed by all the right lines con- 

 necting corresponding points of two collinear planes in space. 

 The focal surface of the congruence is a developable surface of the 

 3. class, and its edge of regression a curve in space of the 3. order. 

 Any two osculating planes of this curve intersect the surface in 

 two conies which are tangent to their line of intersection. Desig- 



