304 THEORY OF COLLINEATIONS. 
In the second case, let Al’, Fig. 37, be the line at infinity, 
and let the triangle PQR be transformed into P,QR, Q and 
R being two invariant points on /’’. The cross-ratio of the 
pencil through A is k = 7 where p, and p are perpendiculars 
from P, and Poni”. Hence we have R= ~*~ =" =k. 
Fig. 38. 
361. Type V. In the first case, when the line / is the line 
at infinity, the triangle PQR is transformed into P,Q,R,. 
But PP, QQ) and PR, are all parallel;-and PP; —@@) — fiz 
for the one-dimensional transformations along PA, QA, RA 
are all translations of equal length. Hence Rk = oe = 
and all areas are unaltered. 
P Po 
Fic. 39. 
In the second case, Fig. 39, when the point A is at infinity 
and / passes through finite space, the triangle PQR is trans- 
formed into P,QR where Q and R are two invariant points on 
