561 
[77.] As verifications, the three right lines 60’, ec’, dd’ concur in the 
point c; bd’, ce’, db’, ine; aa”, bb", dd”, in a’; and aa’, b’d”, db’, in 
a point P;, namely in (411): the existence of which four concwrrences of 
lines was to be expected, from a known principle of homography, as 
consequences of the harmonic relations [76. |. Itis worth noticmg, how- 
ever, how simply these concurrences are here expressed, by the ternary 
symbols of the points, according to the Jaw (18); or, if we choose, by the 
corresponding symbols of the /ines, with the analogous law (25): for 
example, the three last concurrent lines, aa", &e., have for their respec- 
tive symbols, [122], [011], and [115] = [122] + [033]. 
[78.] To examine, in like manner, the analogous relations of ar- 
rangement, on the two new typical lines [ 60. ], namely [021 ] and [021], 
whereof each connects the given point a with four points of second con- 
struction, let us write as eight new temporary symbols of the literal 
kind, more convenient than the former symbols, c” 4, 4," c™ B™ A,” 
e/” a,™, the following : 
6 =(112), ¢=(012), d@=(112), e=(312); 
B=(112), y=(012), 6=(112), «= (312); 
so that the two lines in question are, 
Abede, and aBryée. 
We have thus the eight following new symbolical relations, a being 
still = (100): | 
(4) - (e) = (8), 4) +()=@s (@)-()=2@), ©) + = 4); 
(y) - (4)=(8), (v) + (4)= (i (©) +(B) = 20) (©) - = 4); 
whence result at once the four harmonic relations, 
(sbed) = (abde) = (ABy6) = (aBee) = - 1. 
These two lines from a are therefore homographically divided, the point 
A corresponding to ztse/f, and 6 to B, &c.; and accordingly the four right 
lines, bB, cy, dé, ee, which connect corresponding points, concur in one 
common point, which is easily found to be 8s. And other verdfications, 
by such concurrences, can be assigned with little trouble. 
[79.]. It may assist the conception of the common law of arrange- 
ment, of the five points on each of the two typical lines last considered, 
to suppose that the joining line 08 is thrown off, by projection, to infi- 
nity : or, what comes to the same thing, that the two points 6 and B, 
themselves, are thus made infinitely distant. For thus the harmonic 
equations [78.] will simply express that, im this projected state of the 
Jigure, the four points, d, e, 6, €, bisect respectively the fowr intervals, 
Ac, Ad, Ay, Ad; whence it is easy to construct a diagram, not necessary 
here to be exhibited. The consideration of the éwo other lines through 
the same given point a, which have [012] [012] for their symbols, and 
R. I. A. PROC.—VOL. VII. 41 
