568 
[96.] Itmay be added that, on either the fifth or the sixth trace, the 
two points which we have called first and fourth, are the double points of 
a new involution, determined by the two pairs, second and third, fifth 
and sixth; or, with the recent notations [94.], that aa are the foci of 
the involution dc, By; because the three last harmonic equations conduct 
to this fourth equation, 
(Baya) =~ 1. 
[97.] And, as regards the homography of the divisions on the same 
two traces, if we denote, for the sake of distinction, the six points on 
the sixth trace by a’ .. y’, then (because a! = a) the five lines aa’, bd’, 
cc’, BB’, yy’, or (comp. [ 58. ]) the five lines, 
AD, BoE’, CoC", ByyoRe Coo aa 
ought to concur in some one point: which accordingly it is easy to see 
that they do, namely in the point a’ ; in fact, with the recent significa- 
tion of a, . . and a’, . ., we have the symbolic equations, 
(a’) - (a) =) - (6) = (¢’) = (¢) = (011) = (4); 
(B') - (B) = (9) = (x) = (022) = 2(4'). 
[98.] The two sets of six points, on these two traces, with one point 
common, are thus the points in which a certain six-rayed pencil, with 
a’ for vertex, is cut by the two traces as transversals ; the symbols of the 
six rays being the following : , 
a/ap, = [011]; a’ByB’ = [211]; a’c,c,” = [211]; 
a’a” = [100]; ’3,%s, = [111]; a/c™o"# = [111]. 
And from a mere inspection of thesersymbols, we can infer (comp. 
(83)) that the first and fourth rays are the common harmonic conjugates 
of the two pairs, second and third, fifth and sixth; or that they are the 
double rays of the involution, which those two pairs of rays determine : 
the theorem [96.] being thus, in a new way, confirmed. 
[99.] We have now discussed the arrangements of the poznts on 
those nine typical lines A;, whereof each passes through not less than 
four, nor more than six, of the 52 points in the plane anc; but we have 
still three other typical lines to consider, namely the lines A, and A,, 
of which each passes through at least seven points. ‘Taking first, for this 
purpose, the typical line A,,,, namely, aa’, which contains only seven 
points, whereof the ternary symbols have been assigned in [55. ], and 
the literal symbols there given may be retained, we shall, for the mo- 
ment, reserve the consideration of the two points P,,,; but shall intro- 
duce a new and auxiliary point P;,, on the same line, which may be thus 
denoted : 
an 
A* = (122) = aa/Be!//-cp/” ; 
and which may be said to represent, or typify, a first group of third con- 
