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



To pass now to the inverse of our transformation (3), we 

 note that 



81 T'—8,X'—i =— P n (x'—a)—p*e(y'—&)—i 

 a 2 X' 2 +XX' r'+rn T'* = p 2 [a (x'— a y + /3(x'—a)(y'—l>)-\- 



7 (y'-m 



&2 v'_ a 1 y _ 



8 2 Si 



( v '_ a ) J a8q x q 2 — ae (fa q % -f q x fa 2 ) -f (/3 e — 7 8) ^ x ^ 2 "1 

 _|_(y_£) J ( a e ~ g g ) ft ?a +7 g (ji ?2 + q\ fa) — ye fa fa) \ 



Subtracting 1 from the last expression on the right, we 

 have L. written in x',y. Taking for convenience p — 1, we 

 have for Min terms of x',y' : M'= — 8 (x — a) — e (y — b) — 1 . 

 If now we denote 



C = a(x — af-{-/3(x — a)(y — &)+y(y — bf— 1, 



our transformation Z'and its inverse may be written 1 



(3) T: x'-g= (*- a ) L , y >- b= (y- b ) L 



K6) C — M ' y C—M 



, v ™_, ' (■* — a) M , 7 (y — 3) M 

 (31) T . x — a — ± 1 , y — b = ^~ 1 . 



V ^ C — L ' y C — L 



The base conic (every point of which T and T~ l leave 

 fixed) has the equation 



C — M— L = o. 



The line joining O to the intersection K of A D with 

 BC\s 



M=L. 

 An arbitrary straight line 



(y — b) — m (x- — a) -f- 1 

 is transformed by Z"into the conic 



M\ (y — b) — w(x — a) )■ =l(C — L). 



1 The symnietry of the formuUe suggests that we give initially two pairs of points to 

 fix L=o, M = o (introducing eight parameters) and a base conic through these four 

 points (introducing a ninth parameter). 



