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KANSAS UNIVERSITY QUARTERLY. 



above cases are of great importance in the theory of groups, we 

 state them again in a theorem: 



TJicoron 4. Two conies witli a contact of the second or tttird order 

 determine a perspective collineation witli the co//i/non taiit^ent at the 

 point of contact as its axis and a finite point in the axis as its centre. * 



The centre is, of course, in both cases determined by the two 

 conies and is found as indicated above. In fact the two cases are 

 alike and thus it was not necessary to distinguish them in the 

 above theorem. 



§3. Number and 



[n variant Properties of Perspective 

 Collineation. 



From the preceding development it is known that each two conies 

 tangent to each other determine a perspective collineation with the 

 common tangent at their point of tangency as the axis and the 

 intersection-point of their two other common tangents as the centre 

 of collineation. As there are oo3 conies tangent to a straight line 

 at a certain point, }4 oo3(oo3 — i)=oo6 pairs of conies tangent to 

 each other and to the line at that point can be formed, hence, just 

 as many coUineations. But there are discrete groups among these 

 pairs which represent the same collineation. For, as a perspective 

 collineation is also determined by the centre, the axis, and two 

 corresponding points, it may be represented oo3 times by pairs out 

 of those 006. Hence, there are 006 : oo3=oo3 pairs representing 

 different perspective coUineations. Taking all the points of the 

 axis as points of tangency for pairs of conies determining a per- 



*Our theorem involves the fifth type of Lie's table, pa?c Gfi. '•Continuierliche 

 Gruppen." 



