e 
543 
[19.] As other examples, the fowr planes, 
A,ByCj, AgBC2,  A‘B\C), A’ aB/2B’2, (83) 
have for their quinary equations, 
L+Y+3=Qw+y, C+Y+8=W4+2W, et+yt+stv=4u, 
L2t+yt+st+w=40, (84) 
and for their quinary symbols, 
eo PTS ye Pia nh (85) 
they have therefore a common trace, namely the line 
[Ltis, or arn't, (86) 
because, by (49) and (54), we may now write, 
a’=(011), B’=(101), oe” =(110), (87) 
and the coordinates of each of these three last points satisfy the equation, 
2tyte2=0. (88) 
Accordingly, because we have, by (60) (61) (62) (63), the three following 
sets of symbolical equations of the form (72), 
(a”) = Gi) — (i) = (Be) — (2) = @1) — (C1) = (82) - (C2), 
(BY) = (er) — Ci) = (62) = (4s) = (0) — (4) = (C2) — (44), ¢ (89) 
(0") = (Ai) — G1) = (2) — G) = (41) - G1) = (42) - 2), 
we see that the point a” is the common trace of the four lines, B,c,, BoC., 
BC’), B/c’,; B/’ of O,Aj, Code, C1A4, C/,A’n; and c” of 4B), AlBo, AB’, A/B/o, 
[20.] In all such cases as these, in which we have to consider a set 
of three points P, or a set of three planes I, of which the first is geometri- 
cally derwed from aBcvE according to the same rule of construction, as that 
according to which the second is derived from BcapE, and the ¢hird from 
_ CABDE, we can symbolically derive the second from the first, and in like 
manner the third from the second, (or again the first from the third,) by 
writing, in each case, the third, first, and second coefficients, or coordi- 
nates, in the places of the first, second, and third, respectively. In symbols, 
we may express this law of successive derivation, of certain syntypical 
points or planes [9.] from one ‘another, by the formule, 
if P(aBc) = (xyzwv), then P(BcA) = (zaywv), and P(caB) = (yzawv); (90) 
and if 
T(azc) = [/mnrs], then I(sca) = [nlmrs], and U(cas) = [mnlrs]; (91) 
as has been already exemplified in the systems (27), (60), (61), (62), 
(63), (77), (81), (87), for points or planes, and in (82) for lines, con- 
sidered as traces of planes. In all these cases, therefore, we can, with 
perfect clearness and definiteness of signification, abridge the notation, by 
