M.-P.-Vol. I.] DICKSON— QUADRATIC TRANSFORMATIONS. 15 



3. An elegant geometric construction for T follows 

 readily from (3) as remarked recently by Prof. Haskell. 1 



Thus ^2(^3 — p~^i) = o, X.i (X 3 — o- X 2 ) = 



are transformed by (3) into respectively 



I — pK = o, t; — <r^=o. 

 Hence if we set up two projective pencils 

 \ 3 — p\ x = o, £ — pK= o 



by means of a conic, and similarly for the second pair of 

 pencils, we find for an arbitrary point, determined as the 

 intersection of the rays X 8 — p'X l -=o, \ 3 — a'\. i =o, the 

 transformed point by taking the intersection of the trans- 

 formed rays £ — p £= o, t) — a' £ = o. 



4. The normal type of transformation Q transforms 

 a%-\-bii-\-c%=o into ar) £ -\- b £ £ -\- c Z V "=°- 



In particular, the line b ?/ -f- c £ = o through a vertex is 

 transformed into the line-pair ^ = o,b^-\-cr) = o, the lat- 

 ter evidently making with the sides 77 = o, £ = o angles equal 

 to those made by the given line. The vertex X = V = o is 

 transformed into £ = o, the side opposite. Neglecting the 

 vertices, every bisector of an interior or exterior angle of 

 the triangle is a self-corresponding line. 



If we make the construction on a sphere, we have a sim- 

 ple division into forty-eight compartments by the sides and 

 bisectors. Each of the four centers of the inscribed and 

 escribed circles of the triangle of reference and each of 

 the four analogous points for the symmetrical triangle is a 

 vertex for six triangular compartments. These eight cen- 

 ters are the fixed points under Q, which produces a " pro- 

 jective rotation" through 180° of the six compartments 

 about each center. 



5. An immediate generalization of the transformation Q 

 is obtained by making a correspondence between any point 



JProc. Cal. Acad. Sci., 3d Ser., Math. -Physics, Vol. I. 



