542 
[17.] We may also occasionally denote a point in the given plane of 
A, B, c by the ternary symbol, 
(a, y, 2), or (ays); (74) 
considered here as an abridgment of the guinary symbol (xyz00); and 
the right line which is the trace on that plane, of any other plane I, or 
[¢mnrs | (23), may be denoted by this other ternary symbol, 
[t, m, n], or. [dmn]; (75) 
these two last ternary symbols being connected by the relation, \ 
le + my + nz = 0, (76) 
if the point (ayz) be on the line [lmn]. And the point Pin which any other 
line A, noé situated in the plane axc, intersects that plane, may be said to 
be the trace of that line. 
[18.] For example, the point D, is, by (64), the trace of the line pz; 
and if we write, 
49=(111), B)=(111), = (111), NGA) 
then these three points are the respective traces of the three lines 4, Ag, 
B,By, C,C,; because they are, by the notation (74), in the given plane, and 
we have, by (60) and (61), the three following symbolical equations 
of the form (72), 
(40) + (Ar) + (2) = (Bo) + (Bi) + (Be) = (Co) + (x) + (C2) = (0), (78) 
which express the three collineations, 4944,, BoByBo, Cy0,C,- 
We have also the three other collineations, ap,4’, BD,B’, cp,c’, because 
the quinary symbols (27) (49) (54) (60) give the equations, 
(4) + 4) +@)=@)+@)+@)=©+()+@)=(); (79) 
and these three lines, aa'D,, &c., are the traces of the three planes avn, 
BDE, CDE, of which planes the respective equations (21), and guinary 
symbols (23), are 
y-2=0, s-2=0, g-y=0, (80) 
and [01100], [10100], [11000]; (81) 
so that the ternary symbols of the three last dines, regarded as their traces, 
are simply, by (75), 
[Oli], [101], [110]. (82) 
Accordingly, whether we consider the point a = (100), or a’= (011), or 
D, = (111), (this ternary symbol of D, being congruent to the former gui- 
nary symbol (00011) for that point (60),) we have in each case,the re- 
lation y — z = 0 between its coordinates; and similarly for the two other 
lines. 
