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KANSAS UNIVERSITY QUARTERLY. 



triangle of T is found by putting Xj, yj, and z^ equal to x, y, and 

 z respectively. We thus find that the sides of the invariant tri- 

 angle are x^o, y — z=o, and y-|-z=:0. Solving these equations we 

 find the vertices of the invariant triangle to be A=(o, i, — i), 

 B = (i,o,o), C=(o, i,i). Thus one vertex, A, of the invariant tri- 

 angle is a point of inflection and the opposite side is its harmonic 

 polar. One vertex of the inflectional triangle, xyz=o, lies on this 

 harmonic polar, viz: (i,o,o); the opposite side, x=o, completes 

 the invariant triangle. The position of the invariant triangle is 

 thus completely determined. 



The cross-ratios along the sides AB, BC, CA of the invariant 

 triangle are respectively — a*, a, — i. This may be verified by 

 writing down the cross-ratios of the first six powers of T, assuming 

 T to be given by —a 



(i6) 



T^ and T* are thus shown to be transformations of type IV and 

 order 3, having an identical transformation along the side CA. 

 T^ is of type IV and order 2, having an identical transformation 

 along BC. 



The transformation T may be written down by means of formulas 

 (10) as follows: 



px, 



pZ,: 



