68 KANSAS UNIVERSITY QUARTERLY. 



points A, B, C From these co - conies one can formco^ pairs of 

 conies ; out of these co ^ pairs of conies x) '^- pairs will give rise to the 

 same coUineation. For let us choose any conic L passing through 

 A, B, C. The transformation ^constructed by A' and A\ transforms 

 L into Zj, intersecting L in A, B, C, and V. The course of reason- 

 ing used above shows that corresponding j^oints on L and L^ are col- 

 linear with V. Hence the two conies L and Z^, may be used to 

 construct the collineation Tin the same way that K, K^, and S were 

 used. It is evident that to each of the oo ■■■' conies through A, B, and 

 C there is a corresponding conic, and hence there are oo - different 

 constructions of the same transformation T. 



Theorem 4. A collineation T can he constructed hy means of a 

 pair of intersecting conies in oo- different ivays. 



(3. Collineation Constructed by Means of Two Circles. The 

 collineation constructed by means of two intersecting circles A'and K\ 

 is an interesting special case. Without giving the proofs I shall 

 state only the results, leaving the reader to work out the details. 

 This special collineation transforms angles into equal angles and 

 parallel lines into parallel lines ; it also increases or diminishes all 

 areas by a constant ratio, which is equal to the ratio of the radii of 

 the tw^o circles. In case the circles are of equal radii the collineation 

 is equivalent to a rigid motion of the plane into itself. 



B. — Collineations in vSpace. 



7. Two Intersecting Space Cubics. Let us suppose that a given 

 collineation in space transforms a point ,6' into xSi and *S'] into S-i. The 

 bundle of rays through -S' is transformed into the bundle through aS'i. 

 The two bundles iS) and {Si) are protectively related and hence 

 (Reye's Geometrie der Lage, Zweite Abtheilung, S. 87.) the locus of 

 intersection of those corresponding rays which do intersect is a twisted 

 cubic C passing through both S and S\. In like manner the original 

 bundle through S\ is transformed into the bundle through .SV, and the 

 corresponding intersecting rays generate a second cubic Ci passing 

 through 'S'l and S-i. Since the bundles through xS and *S'i are trans- 

 formed into the bundles through S\ and Si respectively, it follows that 

 the cubic 6' is transformed into the cubic C\. 



The line SSi is a common ray of the two bundles through xS'and S\\ 

 considered as a ray of the bundle through -S it is transformed into the 

 tangent to Cat -S'l; l)ut considered as a ray of the bundle through *S'i 

 it is transformed into the line SiS.,. Hence the tangent to Cat Si 

 intersects Ci, and S-i is this point of intersection. 



Consider a point P on C and its corresponding point Pi on C\. 

 The lines A'Z* and SiP Q.xe corresponding lines in the two bundles 

 through 6" and Si, since they meet on C. But N is transformed into 



