ON THE CONSTRUCTION OF COLLINEATIONS. 



BY H. B. NEWSON. 



1. Introduction. On page 137 of Reye's Geometrie der Lage 

 ( Holgale's translation), is found the following pair of dualistic theo- 

 rems : 



Two conies which lie in the same 

 plane and have one point S in common 

 are correlated projectively to each other 

 if those points of the curves are made 

 to correspond, which lie in a straight 

 line with S. Every common point of 

 the curves different from S is a self- 

 corresponding point. The point S is 

 likewise a self-corresponding point if 

 the curves have a common tangent in 

 this point, /. e., if they touch each other 

 at S. 



Two conies which lie in the same 

 plane and have a common tangent s 

 are correlated projectively to each other 

 if those tangents to the two curves are 

 made to correspond, which intersect in 

 s. Every common tangent to the two 

 curves different from .s- is a .self-corre- 

 sponding line. The line .s itself is self- 

 corresponding only if the curves have a 

 common point of contact in s. 



In vol. IV, page 2-43, of this journal. I deduced the converse of the 

 theorem on the right, and showed how by its use to construct the five 

 types of collineations in the plane. In the Annals of Mathematics, 

 vol. XI, page 148, Prof. Arnold Emch deduced both theorems from 

 the properties of the congruences (1, 3) and (3, 1) of lines in space. 



The object of this paper is to construct the five types of collinea- 

 tions by means of the theorem on the left, and to develop the analo- 

 gous method for constructing the thirteen types of collineations in 

 space. 



A. — Collineations in the Plane. 



2. Construction by Means of Two Conics. Let there be given 

 two projectively related conics A' and Aj intersecting in a real point 

 S. By making use of the principle that corresjjonding points on K 

 and A^j are collinear with S, we can construct the line g^ correspond- 

 ing to any line g of the plane ; we can also construct the point /^j 

 corresponding to any given point P. 



The line g cuts A' in Q and B: join Q and lito S : these joins cut 

 Ki in Qi and Ai corresponding i^oints to Q and R. The line joining 

 Qi and Ai is the line gi which corresponds to g. If a point P be 

 given, we find Pi by drawing two lines g and g' through A cutting K; 

 find by the above constructioai the corresponding lines gi and g\\ 

 these intersect in Ai, the point which corresponds to A. 



If the line g cuts K in a pair of imaginary points, the construction 

 of gi may be accomplished by choosing two points G and G' on g and 

 constructing their corresponding points Gi and G'l ; these new points 



5— K.U.Qr. A— ix 1 [65]— K.U.Qr.-A is 1— Jan. '00. 



