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



One region, as a, may transform into several, all desig- 

 nated a ' ; calling them individually 1, 2, 3, 4, they transform 

 into respectively 1', 2', 3', 4'. The arrows on a line through 

 O indicate the directions in which its points are trans- 

 formed. 



8. Transformations of period two. 



The division into compartments is simplest when the cor- 

 respondence between every pair of corresponding points is 

 mutual, i. e., when the period of the transformation is 2. 

 Let us find the most general quadratic Cremona transform- 

 ation T of period 2 and having three distinct principal 

 points. The condition that T= S Si Q S~* be its own re- 

 ciprocal is Si Q = Q Si~*. Let 



{p x' = a x -j- b y -f- c z f a x=Ax -\-A'y -\- A" z 



py'=a x -\- b' y -f c z Sr 1 = < <r y = B 'x '-j- B 'y '-f- B" z 

 pz'= ax -j- b"y -\- c" z I a z = Cx -j- C'y + C"z 



For Q Si~* and Si Q we have respectively 



, A A' A" , 1 



\x = — 4- - - -j- - - , etc. ; jjl x ■-= —7 — ; , etc. 



x ' y ' z ax-\-by-\-cz 



The condition that Si Q = Q Sr 1 is thus 



(ax -f- by -f- c;?) (Ay z -f- A' x z -f- A"xy) = (a'^-f ^'_y -f- c'^) 

 X (Byz -f- B' xz -j- B" xy) = (tf",v -f- £"_)/ -f- c"^) (Cjy^ -f 

 Cxz -f- C" xy). Hence 



(1) «^4 r 3^'+cy4"=«'^ + ^^'+c'^=«"C , + 3"C"-fc"C" 



(2) aA'=a'B'=a"C' 

 (4) aA"=a'B"=a"C" 

 (6) bA = b'B = b"C 



(3) c^'=c'^'=c"C; 



(5) bA"=b'B"=b"C"; 

 (7) c^4 =c'B = c" C. 



Case I: a^o. From (2) and (3), B'B'=o; from (4) 

 and (5), C" B"=o. 



(I a ) Suppose first B'^o. Thus B'=o, C"=o. From 

 (2) and (4), ^4'=#, a"C' = o and ./4"— 0, a'=o. Since 

 A'=B'=o, C'^^o. Thus #"= 0. Likewise ^4 ^ 0. 



By (5) ? ( 6 )> (3)' (7) m turn > #' = 0, £ = 0, c" = o, c=<?. 

 Hence 6"i is 



t t t ' 7 ff 



x : y ; z = ax : c z : o y . 



