1 8 CALIFORNIA ACADEMY OF SCIENCES. [Proc. 3D Ser. 



collinear with a given point O: (a,b). We must then have 

 identically in x andjy: 



x — a x — a U — a W 

 W y~—b = y—6 = V—b W 



Hence there must exist a linear function L (x,y) such that 

 (2) U—aW=(x — a)L; V—bW=(y—b)L 



from which, b U — a V= (b x — ay) L. 



For the case in which U=o, V= o , W=o have three 

 distinct points of intersection, these can not all lie on Z.=o, 

 since T would then reduce to a linear transformation. 

 Hence one point of intersection is O: (a,b) and the other 

 two lie on L = o, say A and D. Hence by virtue of (2) 

 our transformation T may be written 



/ v 1 ( x — a ) L , (y — b) L 



The line L = o is transformed into the point (a, b), so that the 

 the latter is not fixed. Hence the fixed points of (3) all lie 

 on the " base conic," 



(4) W=L. 



T contains nine arbitrary constants, depending on the arbi- 

 trary point (a, b), the arbitrary conic through it 



(5) W=a(x — afJ r 0(x — a)(y — b)J r y(y — bf + 



8 (x — a) -f- e (y — b) = o, 



and the arbitrary points A and D on W=o (distinct and 

 different from O) to determine the line L = o. We will fix 

 A and/? as the intersections of W =0 with 



/ 6 x OA: X=f x (x — a)A r qx{y — b)=o 



K) OB: r=fr(x — a)+q 2 {y — b) = o 



solving, ]>(*—«)- q*x-qi r 



Hence (7) W=.a*X % + \Xr-\- a x T* + B % X— l x T=o 



where f«i = a^i 2 — $f\ q\ -f- ypi 2 , &i=p(&oi — efli) 



a 2 = a q 2 2 — fifl 2 q % -f y fi 2 , B 2 = p(8q 2 — ep 2 ) 

 X = /3 (pi q 2 -\--p 2 q x ) — la q x q 2 — 2 7/1^2 



