248 KANSAS UNIVERSITY QUARTERLY. 



invariant points. But if a projective transformation leaves more 

 than two points of a line invariant, it leaves all points on the line 

 invariant, (Cont. Gruppen, page 117). Therefore every point on 

 the line 1 is an invariant point of the transformation. Any line 

 g drawn throngh C intersects 1 in some point as G. Therefore the 

 line g having two points G and C invariant is an invariant line. 

 Thus we see that every line through C is an invariant line. Hence 

 we conclude that the transformation determined by the two conies 

 K and K' touching the line 1 at the same point leavesthe point C, 

 all points of the line 1, and all lines through "C invariant. The 

 two remaining points of intersection of the two conies may be 

 either real or imaginary, the result is the same in either case. If 

 these remaining points of intersection are coincident, i. e. if the 

 two conies have double contact, the resulting transformation is still 

 of the same character and the invariant figure the same. This con- 

 stitutes Case IV. 



When the two conies have a contact of the second order at the 

 point L on the line 1, the invariant figure takes still another form. 

 In this case only one other common tangent x can be drawn to the 

 two conies. This common tangent intersects 1 at A, (Fig. 2 (d) ). 

 The transformation determined by the two conies in this position 

 leaves invariant all points of the line 1 antl all lines through. A. 



If the two conies have contact of the third order at L, then 1 is 

 the only common tangent they have, (Fig. 2 (e) ). Such a trans- 

 formation leaves invariant every point of the line 1 and every line 

 through L. The invariant figure is therefore the same as before. 

 This constitutes Case V. 



We have now enumerated all the special positions which the 

 conies K and K' can take and there are no more cases to be con- 

 sidered. 



TJtcoroii J. Ei'cry projective triDisfonitaiioii of tJic plane deierniiiied 

 by two conies K and A'' touching a fixed line 1 of the plane leaves a cer- 

 tain plane figure invariant. Titer e are five cases to be considered: 



Case I. WJien the two conies are not in contact, the invariant figure 

 is a triangle. 



Case //. When the two conies have contact of the first order not on 

 1, the invariant figure consists of two invariant lines, their point of 

 intersection, and an invariant point on one of the invariant lines. 



Case III. IVhen the two conies have contact of the second order not 

 on 1, the invariant figure consists of one invariant line and one invar- 

 iant point on that line. 



