NEWSON : ON THE CONSTRUCTION' OF COLLIXEATIONS. 67 



through 6' cuts A' and A'l m a pair of corresponding points, it follows 

 that A, B, and C are self-corresponding or invariant points on A' and 

 K\. In other words A, B, and Care invariant points and the lints 

 AB, BC, and CA are self-corresponding or invariant lines of the col- 

 lineation. Hence in the most general case a projective transforma- 

 tion of the i^lane leaves invariant the vertices and sides of a triangle. 



The three points A, B, and C are either all real or one is real and 

 two are conjugate imaginary; for the real conies A' and I\\ intersect 

 either in four real points, in two real and two conjugate imaginary 

 points, or in two pairs of conjugate imaginary points. Since S is 

 real. A, B, and C are either all real or one is real and tvro conjugate 

 imaginary. 



Theorem 2. ^1 real eollineation in its most general form leaves a 

 triangle i7ivariant which is real in all of its parts or has one real 

 and two conjugate imaginary vertices. 



4. Five Types of Collineations. The five well-known ty]3es of 

 collineations in the plane are obtained by taking the tw'o conies K 

 and K\ in different special relations to each other. Thus when the 

 two conies A'and K\ of the above construction intersect in four points 

 we have what is called type I. This type has two sub-types, hyper- 

 bolic and elliptic, according as the invariant triangle is real in all its 

 parts or partly real and partly conjugate imaginary. 



If the two conies have contact of the first order, as for example 

 when A and B coincide, the transformation is said to be of type II. 

 If the two conies intersect at S and have contact of the second order 

 at a point A, the invariant figure is a lineal element A\, and the trans- 

 formation is said to be of type III. 



If the conies A" and K\ have contact of the first order at S. then. 

 the common tangent to A" and K\ at xS is an invariant line; also the 

 lines joining S to the other two points of intersection are invariant lines 

 of the eollineation. Thus we have three invariant lines through 6" 

 and one invariant line not through S. The transformation of this 

 type is a perspective eollineation wath its vertex at S and its axis 

 through the other two points of intersection of A' and A", ; it is said 

 to be of type IV. If A and K^ have contact of second or third order 

 at S, the eollineation is still perspective but with its vertex on its 

 axis ; it is said to be tyi^e V. 



Theorem 3, The construction of a eollineation hy means of two 

 conies K and Ky gives rise to five distinct types of collineations, ac- 

 cording to the mutual position of the two conies. 



5. CO- Coxsteuctions of the Same Collineation. Any given 

 eollineation Tcan be constructed in x- ditferent ways, as w^e proceed 

 to show. There are co "^ conies passing through the three invariant 



