286 
Se Wituram R. Hamitron read a paper— 
ON ANHARMONIC CO-ORDINATES. 
1. Ler ABC be any given triangle; and let 
O, P be any two points in its plane, whereof 
O shall be supposed to be given or constant, 
but P variable. Then, by a well-known the- 
orem, respecting the six segments into which 
the sides are cut by right lines drawn from yg y 
the vertices of a triangle to any common e Fig. 1. 
point the three following anharmonics of 
pencils have a product equal to positive unity :— 
(A.PCOB).(B.PAOC).(C. PBOA)=+1. 
It is, therefore, allowed to establish the following system of three 
equations, of which any one is a consequence of the other two :— 
oa(A. PCOB); == (B.PAOC); 7 (C.PBOA); 
- and, when this is done, I call the three quantities x, y, 2, or any quan- 
tities proportional to them, the Anharmonie Co-ordinates of the Point P, 
with respect to the given triangle ABC, and to the given pot O, And 
I denote that point P by the symbol, 
P=(a, y, 2); or, P= (ta, ty, tz); &e. 
2. When the variable point P takes the given position O, the three 
anharmonics of pencils above mentioned become each equal to unity ; 
so that we may write then, 
e=y=z=1. 
The given point O is therefore denoted by the symbol, 
O— (514); 
on which account I call it the Unit-Point. 
3. When the variable point P comes to coincide with the given 
point A, so as to be at the vertex of the first pencil, but on the second ray 
of the second pencil, and on the fourth ray of the third, without being 
at the vertex of either of the two latter pencils, then the first anhar- 
monic becomes indeterminate, but the second is equal to zero, and the 
third is infinite. We are, therefore, to consider y and z, but not a, as 
vanishing for this position of P; and consequently may write, 
A=(1, 0, 0). 
In like manner, 
B=(0, 1, 0), and C =(0, 0, 1); 
and on account of these simple representations of its three corners, I call 
the given triangle ABC the Unit- Triangle. 
