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



and those of the inverse transformation will be 



X : fjb : v = v ( \' -j- fjJ ) : — p! 2 : — p! 1/. 



§ 3. Transformations in the general form. 



Suppose the formulae of transformation to be given in 

 the general form 



p y l = <p ( Xi , X 2 , x 3 ) = d\ Xi 2 -f h x£ + Ci x 3 2 -f 



2f X 2 X 3 + 2 ^1 #3 «i 4- 2^i X\ X% 



p y 2 = yjr (xt, x 2 , x 3 ) = a 2 X\ -f- b 2 x 2 2 -f c 2 x 3 2 -f- 

 (15) 2f 2 X 2 X 3 -\- 2g 2 X 3 X X -f 2 ^ 2 *#1 X 2 



P )'3 = X ( *1 > ^2 , #3 ) = «3 *i 2 -1- £3 * 2 2 -f" C 3 .V 3 2 -f 

 2/3 "^2 Xz -f 2 £3 ^3 #1 -f 2 /Z 3 iVl ^2 



It is then in given cases frequently very difficult to trans- 

 mute these formulae into the reduced form (3), even when 

 we know that they are really birational. The Jacobian of 

 $ , yjr , x must have a double point at each of their intersec- 

 tions, and must therefore degenerate into the triangle of 

 which these intersections are the vertices; that is to say, 

 the equation of the Jacobian will be 



The test that the Jacobian shall so degenerate, is that its 

 Hessian shall do likewise, so that the coefficients of the 

 Jacobian and of its Hessian must be proportional. But, 

 supposing this test to be satisfied, it is usually by no means 

 easy to factor the Jacobian. The formulae of the inverse 

 transformation can however be established in another way, 

 without actually finding the coefficients of the reduced form 

 (3), and to accomplish this is the object of the present 

 paper. 



If we write X = o in equations (3), we find 



y\ : Ji ' ys = p\ ' P2 '• pz • 

 Similarly, p, = o gives 



y\ : V2 : ys = qi : qi  qz 



and v = o gives 



y\ : y% ' ys = n : r 2 : r 3 . 



