l6 CALIFORNIA ACADEMY OF SCIENCES. [Proc. 3D Ser. 



P and the intersection P' of the straight lines harmonically 

 separated from P by each pair of opposite sides of a com- 

 plete quadrilateral A C B D. Taking its diagonal triangle 

 G E F as our triangle of reference, viz. : 



GF=x = o, F E=y = o , E G = z = o , 



the vertices of the quadrilateral may be defined by 



A (a:b:c), B{ — a:b:c), D(a:b: — c), C (a: — b:c). 



The line G P' separated from P (xi : yi : z\) and hence from 

 G P, viz . , 



X\ 



x — — z = o , harmonically by 



a a 



G D \ x 4- - - z = o , G A: x — - - z — o 

 1 c c 



a' 2 z\ 

 has the equation x — —t, • z = o . 



u C X\ 



a 2 y\ 

 Similarly for F P : x — y^ — ' y -= o . 



Hence P' has the coordinates 



a 



b 2 c 2 



xi yi z\ 



The only fixed points are the four vertices. If one of these 

 should be the unit point 1:1:1, the transformation is Q. 



6. Another normal type of quadratic Cremona trans- 

 formation 1 is defined geometrically by keeping fixed one 

 pair of opposite sides of a hexagon inscribed in a given 

 "base conic' and making correspond the two variable 

 points of intersection of the other two pairs of opposite 

 sides. Let the fixed pair, A C and D B, intersect in the 

 point O, and take as triangle of reference : 



D = % = o, A = v = o, A &=Z = o. 

 and as base conic a%v-\-&Z£- i r c v£ J r d £ 2 = o. 



^h. K. Dickson, Rendiconti del Circolo Matematico di Palermo, Vol. IX, 1895; also, 

 American Mathematical Monthly, 1895. 



