NEWSON : ON THE CONSTRUCTION OF COLLINEATIONS. fiU 



Si and P into J\; hence the line SJ^ is transformed into <S'iiA. 

 Therefore SiP and Si Pi must coincide, since they each correspond to 

 SP. Hence P and Pi, corresponding points on the two cubics C and 

 Ci, are coUinear with Si. From this it is evident that the two cubics 

 C and Ci lie on the same quadric cone with vertex at Si. 



Theorem 5. A collineation I' in space which transforms a point 

 S into Si and Si into Si transforms the cubic C into Ci; C and Ci 

 intersect in Si and lie on the same quadric cone with vertex at Si; 

 a pair of corresponding points on C and Ci are coUinear with Si. 



8. CONSTRUCTIOK OF A COLLINEATION BY MeANS OF C AND C,. By 



making use of this last tlieoi'em we can construct the plane pi corre- 

 sponding to any given plane p. The plane /) cuts C in three points 

 P, P', P" : join these three points to .Vj : these joins cut C^ in P^, 

 P\, P" I, corresponding points to P, P' , P" . Pass a plane through 

 i^j, P\, P\ and this is /Jj, the plane which corresponds io p. 



To construct the line g^ corresponding to any given line g in 

 space, pass two planes through g and by the above method construct 

 their corresponding planes; these intersect in g\ the required line. 

 To construct the point Pi corresponding to any given point P in 

 space, pass three planes through P and construct their corresponding 

 planes; these new planes intersect in Pi. 



If a plane p cuts C in one real and two conjugate imaginary points, 

 only the point corresponding to the real point of intersection can be 

 constructed. In this case we can choose two other real points in p 

 and construct their corresponding points ; thus pi is determined. 



If the plane p is tangent to G, the corresponding plane jfi will be 

 tangent to C\ and the two points of contact will be coUinear with Si. 

 If the given plane passes through 'S'l and cuts C in P and P\ the cor- 

 resi^onding plane pn will i)ass through P\ and P'l and the point where 

 the tangent to C at S\ cuts C\, i. e., at S-i. 



Theorem 6. A projective tranfovmation of space can he completely 

 constructed hy means of two cubics C and Ci i?itersecting in a point 

 Si and lying on the same quadric cone with vertex at Si. 



9. Invariant Points, Lines, and Planes. Two cubics C and C'l 

 lying on the same cone A' intersect at its vertex and generally in four 

 other points A, B, C. D. This may be shown as follows : Two twisted 

 quartics on the same surface F of the second order intersect in eight 

 points. When /^degenerates into a cone, each quartic degenerates 

 into a cubic and an element of the cone. The vertex of the cone is a 

 double point on each degenerate quartic and hence counts for two in- 

 tersections. The points in which the elements belonging to each 

 cubic intersect the other cubic count for two more points of intersec- 



