12 KANSAS UNIVERSITY QUARTERLY. 



(/). The two conies K and K* may be represented by degenerate 

 parabolas, i. e., by straight lines which coincide. Centre and axis 

 of collineation are at infinity, and as they are coincident it follows 

 from k=( OD CO AA^) k=+i- The two systems are therefore simi- 

 lar and in similar position; in the relation of dilation and of 

 corresponding equal areas. Hence they are congruent. How 

 the construction of corresponding points is made is seen from 

 Fig. 7. Let V CO and V^ 00 represent the two coaxial parabolas. 



To a point A the corresponding A^ is found by drawing AV and 



00 

 AC parallel to VVi (the two tangents to the conic VV^) and 



intersecting AC by the parallel to AV through V^. (AV and AC 



intersect the -axis of collineation, or the line at infinity in two 



points. The tangents from this point to the conic V^ c» are A^V^ 



and A^C). We add this construction here in order to show that 



also in the case of singularities the construction is applicable. 



We have now seen that all the common cases of perspective 



collineation are expressed by the characteristic constant k together 



with the positions of the centre and axis of colliixeation, or result 



from certain positions and relations of the two conies. What 



remains 3'et to consider are the so-called pseiido perspective collinea- 



tions. * 



Fig. 8. 



Here to one point may correspond a whole system of points and 

 vice-versa. These singularities can be classified according to 



*Prof. Newson introduced the term pseudo transformation into geometry. Singular 

 perspective collineations can therefore also be called jiseudo perspe<;tive collineatiuns. 



