244 KANSAS UNIVERSITY QUARTICRLN . 



exactly as before by considering ranges of four points each on a^, 

 j3^, y^, 8' and constructing the corresponding ranges on u, /i, y, 8. 

 As before we shall have four points on 1 in tt^ and their four corre- 

 sponding points on 1^ in tt: the lines joining these corresponding 

 points all lie in the plane tt and together with 1 and 1, determine a 

 conic K in tt, every tangent to which cuts 1 and Ij in corresponding 

 points. Let the points on 1, corresponding to A, B, C, D on 1 be 

 Aj, Bj, C,, D,; and let the lines joining AAj, BBj, CC^, DD^, 

 be a, b, c, d. Now a and a^ are corresponding lines of the two 

 planes. For A and Aj are two points of tt whose corresponding 

 points in tt' are A and A'; hence a'^, the join of A and A^, corre- 

 sponds to a, the join of A and A,. For the same reason b and bS 

 c and c^, d and d', are corresponding lines of the two planes. 

 Thus we have in general: 



The taiii^oifs to iJic tioo coiiics K and A' from aiix point oit 1 arc corre- 

 sponding lines of the two planes. 



By the aid of the conies K and K' we can now construct the point 

 pi in TT^ corresponding to any point 1' in tt. Draw the tangents 

 from P to K and let them cut 1 in Q and R; the lines in tt^ corre- 

 sponding to these tangents from P are the tangents from Q and R 

 to Ki; these meet in P^, the point corresponding to P. 



If we have given any line as g in tt, we can construct the corre- 

 sponding line g' in tt' as follows: take two points P, and Pg on g 

 and in the manner just explained construct the corresponding 

 points Pj and P.]; the line g> joining P{ and Pi corresponds in tt^ 

 to g in TT. 



Since the constructions in the two planes are exactly alike, these 

 operations are strictly reversible. By means of the conies K and 

 Ri, the configuration in tt corresponding to any given configuration 

 in tt' can be constructed. And further, one of the planes as tt^ may 

 be revolved about 1 until it coincides with tt, and then the construc- 

 tion thought of as all in the same plane. We shall make constant 

 use of this last conception. 



The lines joining corresponding points of the planes tt and tt' 

 form a linear congruence (Strahlen-Congruenz) of the third order 

 and first class; (see Reye's Geometrie der Lage, II. Band, p. 94). 

 By means of this congruence the projection of tt on tt^ is completely 

 determined. By a revolution about 1 this new set of points may be 

 brought back to tt and both sets of points thought of as existing in 

 the same plane. This operation of projecting the points of tt into 

 a new system of points on tt^ and by a revolution about 1 bringing 

 the new system back to tt will be called a Projective Transformation 



