l6 KANSAS UNIVERSITY QUARTERLY. 



To this line-element all the perspective coUineations can be 

 constructed which contain it as an invariant element. This can be 

 done in two ways, first by taking the point as the centre and the 

 • line through it as an invariant ray of the perspective coUineation, 

 second by assuming the point as a point of the axis and the ray 

 through it as a ray through the indeterminate centre of the coUine- 

 ation. By the same reasoning as before we find that in each case 

 oo3 perspective coUineations have a line-element in common. The 

 two cases are in a dualistic relation and may be expressed in the 



Til CO re III J. There are ^"^perspective coUineations leaving a line- 

 eleiiiciit invariant. 



Combining a line-element with either the points of a straight 

 line or the rays through a point as the invariant figure, two other 

 cases are obtained. If the points of a straight line and another 

 straight line through one of these points are invariant, the centre 

 of coUineation may be any point on that other invariant line. This 

 gives c»i centres, and one on the same line as the axis of collinea- 

 tion. Adding to each centre two corresponding points, or what is 

 the same, a characteristic anharmonic ratio, the number of per- 

 spective coUineations is multiplied by coi, so that W- perspec- 

 tive coUineations satisfy the given conditions. Hence the 



Theorem 8. There are ooS perspective coUineations leaving the points 

 of a straight line and another straight line invariant. 



In the dualistic case, where the rays through a point and an- 

 other point on a ray through the first point are invariant, the axis 

 of coUineation may be any straight line through that other invari- 

 ant point. This gives odI axes and one and the same point as the 

 centre of coUineation. 



Adding to each axis two corresponding lines, or, what is the 

 same, a characteristic anharmonic ratio, the number of coUinea- 

 tions is multijplied by ooi, so that coS coUineations satisfy the 

 given conditions. Hence the 



Tlieorcin g. There are oo"- perspective coUineations leaving the rays 

 through a point and another point on one of the rays through the first 

 point invariant. 



Finally there exists a system of coUineations in which all points 

 of a straight line and all rays through a point are invariant. Here, 

 a special coUineation is simply characterized by two corresponding 

 points, or the characteristic anharmonic ratio. There are just coi 

 such coUineations. This result may be stated in the 



Theorem lO. Tliere are ooi perspective coUineations leaving the 

 points of a straight line and the rays through, a point invariant. 



