352 
15. Let any plane II intersect the three given lines pa, DB, pc in 
points @, R, s; and let /mur be any positive or negative numbers, such 
that 
= (04/240), — = (DB’,BR), 5 = (p0',08) ; 
Sis 
then I call 7, m, », 7, or any numbers proportional to them, the Anhar- 
monic Coordinates of the Plane 11; which plane I also denote by the 
Symbol, 
H=[1, m, n, r]. 
In particular the four faces of the unit pyramid come thus to be denoted 
by the symbols, 
gop =[1, 0, 0,0], cap=[0, 1,0, 0], asp = [0, 0, 1,0], anc=[0, 0,01]; 
and the six planes through its edges, and through the unit point, are 
denoted thus :— 
pce =[1, 0, 0,-1]; caz=(0, 1, 0, -1]; azz =(0, 0,1,-1]; 
ape =[0,1,-1, 0]; sp—E=[-1, 0,1,0]; coz=([1, — 1, 0,0]; 
in connexion with which last planes it may be remarked that we have, 
generally, as a consequence of the foregoing definitions, the formule, 
n l m 
ar (Ba/’CL), a (cB//am), Fie: (ac’’BN), 
if t, M, N be the points in which a variable plane TI intersects the sides 
Bc, &c., of the given triangle anc: as we have also, generally, 
; = (AD. CEBP), == (BD. AECP), =# (cD. BEAP). 
16. If a point, Pp = (ayzw), be situated on a plane, 11 =[lmnr], then I 
find that the following relation between their co-ordinates exists, which is 
entirely analogous to that already assigned (6) for the case of a point and 
line in a given plane, and is of fundamental importance in the applica- 
tion of the present Anharmome Method to space: 
le + my +ns+rw=0; 
or in words, “‘the sum of the products of corresponding co-ordinates, of 
point and plane, 7s zero.” 
For example, all planes through the unit point (1, 1, 1, 1) are subject 
to the condition, 
l+m+n+r=0, 
as may be seen for the six planes (15) already drawn through that 
point u; and the six points a’ B/ c/’ a’, B’, c’, (14), in which the six 
edges BC, CA, AB, DA, DB, Dc, of the given or unit pyramid axzcp, intersect 
the six corresponding edges of the inscribed and homologous pyramid 
A,B,C,D,, with the unit point x for their centre of homology, are all ranged 
