NEWSON: projective TRANSFORAfA'JiONS. 



55 



are independent of the character of the coordinates whether rec- 

 tangular or oblique. If oblique coordinates are used then x, and 

 y are proportional to the perpendicular distances from P and P, 

 on Al and Al' some line through A. 



17. Implicit Normal Form of Type III. Let the coordinates of 



the points A,P,P, be (A,B), (x,y) 

 (Xj,y,) and let the line 1, Fig. 7, 

 make with the axis of x an angle 

 whose tangent is p and let Al' make 

 £ with the axis of x an angle whose 

 tangent is p'. Replacing'! the per- 

 pendiculars in the above relation by 

 their analytic expressions. "in terms 

 of the C()ordinates of A, P, P^ and 1 



(16) 



The line AT may be chosen so that 



-(h' + a)a. (i6a) 



z[ ^i without altering 



Pi I 



the generality of the result. 



18. The Explicit Normal Form for Type III. These two equa- 

 tions are linear in x^ and y^ and their solutions for x, and y, give 

 the explicit form for the equations of Type III. Solving we have 



iX y I o 



Ia B I A 



I p o Aaa-f I 



II P, o Al 



{I a-3-h'a) 



'a-\- I 



aS—hV 



