287 
4, Again, let the sides of this given triangle ABC be cut by a given 
transversal A’B’C’, and by a variable 
transversal LMN. Then, by another B 
very well known theorem respecting 
segments, we shall have the relation, 
(LBA'C) . (MCBYA) . (NAC'B) =+1; 
it is therefore permitted to establish 
the three equations, 
~ = (LBA‘C), - = (MCB‘A), -. (NAC'B); 
where /, m,n, or any quantities pro- 
portional to them, are what I call the 
Anharmonic Co-ordinates of the Line 
LMN, with respect to the given triangle ABC, and the given transversal 
A’B'C'. And I denote the line LMN by the symbol, 
LMN =[1, m, n]. 
For example, if this variable line come to coincide with the given line 
A/B/C’, then 
Fig, 2. 
l=m=n; 
so that this given line may be thus denoted, 
APSO = [iid tls 
on which account I call the given transversal A'B’C’ the Unit-Line of 
the Figure. The sides, BC, &c., of the given triangle ABC, take on this 
plan the symbols [1, 0, 0], [0, 1, 0], [0, 0, 1}. 
5. Suppose now that the wnit-point and unct-line are related to each 
other, as being (in a known sense) pole and polar, with respect to the 
given or wnit-triangle ; or, in other words, let the lines OA, OB, OC be 
supposed to meet the sides BC, CA, AB of that given triangle, in points 
Fig. 3. 
ae? B’, 0”, which are, with respect to those sides, the harmonic con- 
jugates of the points A’, B’, C’, in which the same sides are cut by the 
BR. I. A. PROC.—YOL. VII. 27 
