302 THEORY OF COLLINEATIONS. 
In this case R the ratio of areas is equal to k the cross-ratio 
of the transformation of the pencil through A. 
In the second case let the line Al’, Fig. 34, be the line at 
infinity and let the triangle BP@ be transformed into BP,Q,. 
The cross-ratio along BA is k=(Ba QQ) = ae The 
cross-ratio of the pencil through A is also k= A(#~p,p) 
Pi A BP, Q, BQ. Pi , 
=~ Hence we have R= 7 p59 = Ba.» =k. 
Fig. 35. 
359. Type III. In this case the path-curves of the one- 
parameter group G,’( AlS) to which T” belongs are parabolas 
similar and similarly placed; 7. e., coaxial parabolas having 
equal latera recta. Consider the area of the segments of the 
parabola g cut off by tangents to h, Fig. 35. If T” trans- 
forms PQ into P,Q,, then the area cut off from g by PQ is 
transformed into the area cut off by P,Q,. But these areas 
are known to be equal, (Salmon, 396); hence, R = 1. 
