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Kansas university quarterly. 



6 Explicit Normal Forms for Type I. Equations (5) are 

 linear in Xj and yj, and may be solved for these quantities, giving 

 the explicit normal forms of the transformation T. (See Exer- 

 cise i). Solving we get 



(6) 



If we pass from Cartesian to homogeneous coordinates, these 

 forms may be written. 



px^ = 



(7) 



Making the C's and z's unity and dividing the ist and 2nd by 

 the 3rd, we get equations (6). 



The law of formation of these determinants is evident. 

 The determinant of the invariant triangle is bordered above by 

 X, y, z, on the side by A, k^A^, kgA.;,, etc. 



^2. Type II; Us Properties and Normal Form. 



7, Two Cross-Ratios and Two Characteristic Constants. 



We come now to the consideration of the two-dimensional trans- 

 formations of the second 

 type, whose invariant figure 

 is composed of two lines, 

 their point of intersection, 

 and a second point on one 

 of these lines. Let T' be a 

 transformation of this type 

 leaving invariant the real 

 plane figure ABC, Fig. 5. 

 The transformation T' in- 

 duces along the invariant 



Fig, 5. 



