300 THEORY OF COLLINEATIONS. 
357. Value of R, the Ratio of Areas. Since every collinea- 
tion in G,(/~ ) alters all areas in a constant ratio R, it is im-. 
portant to find the value F in terms of the natural parameters - 
of the collineation. Each type of collineation requires sepa- - 
rate treatment. 
Type I. Let us consider a collineation T and its in- 
variant triangle having the side A’A” at infinity. Let PQ, 
Fig. 32, be a tangent to a path-curve C and let it be trans- 
formed into another tangent, P,Q,, to the same path-curve. 
The area APQ is transformed into AP,Q,, and we wish to 
find the ratio of these areas. The cross-ratio along the side 
AA is (Ac? — <p - The cross-ratio along the side 
BiG. 32: 
AVAU Ish — (eA cot) @) — ae, The ratios of the two areas 
are given as follows: 
Area of AP, a AP, . AQ, sin # 
R= lee 
- Area of A P Q - AP. AQ sing 
