206 ON A CORRESPONDENCE OF POINTS [512 
Correspondence, A, B, C, D at P and A', B', C’, D' at P' subtending equal angles. 
6. Consider now in piano the points A, B, C, D which at P, and the points 
A', B', C', D' which at P', subtend equal angles. Let a, b, c, f, g, h denote the 
perpendicular distances of P from the lines BC, CA, AB, AD, BD, CD respectively; 
and the like as to a!, V, c', f, g', h'. Observe that, neglecting constant factors, a, b, c 
are what were before represented by x, y, z\ we may consider the coordinates of P 
in regard to the triangles ABC, BCD, CAD, ABD to be {a, b, c), (a, h, g), (b, f h), 
(c, 9> f) respectively. We have in regard to ABC the point 0 as before, and in 
regard to BCD, CAD, ABD the points 0 1 , 0 2 , 0 3 respectively. Then A, B, C at P 
and A', B', C' at P' subtending equal angles, we may write 
a' : V : c' 
BCO • CAO ' ABO’ 
viz. BCO = 0 is here the circle through B, C, 0, and the like for CAO and ABO, 
the expressions being multiplied into the proper constant factors to take account of 
the constant factors whereby a, b, c and a', b', c' differ from x, y, z and x', y', z 
respectively. 
We have in like manner 
a! : h' : g' = 
BCO 1 • 
CDO ± 
‘ BDO, 
b 
. / 
h 
cao 2 
' ADO a 
■ CD0 2 
c 
9 
. f 
AB0 3 
' BD0 3 ' 
AD0 3