351 
ON ANHARMONIC CO-ORDINATES. 
13. Proceeding to space, let a, B, c, D be the four corners of a given 
triangular pyramid, and let & be any fifth given point, which is not on 
any one of the four faces of that pyramid. Let Pp be any sixth point of 
space; and let xyzw be four positive or negative numbers, such that 
x 3 
= (BC. AEDP), . = (CA. BEDP), ig (AB. CEDP); 
the right-hand member of these equations representing anharmontes of 
pencils of planes, in a way which is easily understood, with the help of 
the definition (5) of the symbol (ancp). Then I call a, y, 2, w (or,any 
numbers proportional to them), the Anharmonie Co-ordinates of the 
Point P, with respect to what may be said to be the Unit-Pyramid, axscn, 
because its corners may (on the present plan) be thus denoted, 
a =(1, 0, 0, 0); B= (0, 1, 0, 0); c=(0, 0, 1, 0); p=(0, 0, 0,1); 
and with respect to that fifth given point =, which may be called the 
Unit-Point, because its symbol, in the present system, may be thus 
written :— 
= = (1,1, 1, 1). 
And I denote the general or variable point by the symbol, 
P = (2, y, 8, w). ’ 
14, When we have thus five given points, a...£, of which no four 
are situated in any common plane, we can connect any two of them bya 
right line, and the three others by a plane, and determine the point in 
which these last intersect each other, deriving in this way a system of 
ten lines, ten planes, and ten points, whereof the latter may be thus de- 
noted :— 
a’=Ber ADE= (0, 1,1, 0), B’/=&c., c’ = &e. ; 
A; = AE’ BopD=(0, 1,1, 1), 3B, =&c., c,= &e.; 
A,=AD: BCE=(1, 0, 0,1), Bo= &., c2= &e.; 
D, = DE’ aBc = (1, 1, 1, 0),; 
and the harmonic conjugates of these last points, with respect to the ten 
given lines on which they are situated, may on the same plan be repre- 
sented by the following symbols :— 
a”=(0,1,-1, 0), B” = &. c’ = &e. 
An, =(2,1, (1,1), 3B, = &e. co, = &e. 
Mo (1, 06-1), no = &e. 0, = Ke. 
vi=(1,1,1, 2); 
so that 
(pa’ca””) =...=(#A,AA)) =...=(DA2Aa2) =... = (ED,DD";) =— 1. 
R. I. A, PROC.—VOL. VIL. 3D 
