194 Proceedings of the Royal Society of Edinburgh. [Sess. 
the new origin x' = 0 be the projection of aj = l. This is effected by the 
transformation 
1 
y 
x -\- 1 ar+ i 
by which straight lines remain straight lines. 
6 ) 
Substituting in (3) and (4) 
we obtain 
1 -\-x 
^= t =! — i 
1 — x 
y = 
2 y’ t 
\—x 
and 
(x — 1 ) 2 = (x + 1 ) 2 sin 2 a + 4 y 1 ’ 2 
tan 2 a . 
• (7) 
• ( 8 ) 
■*: (9) 
respectively. 
(x — 1 ) 2 = {x + 1 ) 2 cosec 2 B — 4 y' 2 sec 2 B . 
