570 
[102.] Introducing now the consideration of the two lately reserved 
points P,,,[99.], of second construction and third group [45.|, upon the 
typical line A,,,, we may derive them from the point 2, the two points 
P,, and the two points P,,., upon that line aa’, by the two following har- 
monic equations : 
(aa!/a/a*) = (45D, A,"") = - 1; 
or by these two others, 
(aa’A a”) = (40, A/"4,") = — I, 
which may indeed be inferred from the two former, with the help of 
the relations between the six points previously considered: for, in ge- 
neral, if abc, a'b'c’ be collinear triads in involution, and if d and d’ be 
the harmonic conjugates of b’and c’, with respect to the two pairs, ab, ac, 
they are also the harmonic conjugates of 6 and c, with respect to the 
two other pairs, ac’, ab’; or in symbols, 
(abc'd) = (acb'd’) = — 1, if (ab/bd) = (ac'cd’) = - 1, 
when the three harmonic equations [81.] exist. We have also, gene- 
rally, under these conditions, the equation 
(ada’d’) =-1; 
for example, on the line aa’, we have 
(Gaara) == 1 
[103.] It is scarcely worth while to remark that the 15 lines A,,, of 
the net, as being all syntypical, are all homographically divided ; although 
it may just be noticed, as a verification, that the six lines, 
Mat ULL ivi iv, iv 
BC, B/C’, BC”, Boy, BC”, B, Cy’, 
which connect corresponding points on the two other lines of the same 
group in the given plane, namely BB'p, and cc’D,, concur in one point 
a”. But it may not be without interest to observe, that a* is the com- 
mon harmonic conjugate of a, with respect to each of the three pairs, 
A’D,, A’/’Ay, A”Ay”; which three pairs,* or segments, form thus an invo- 
lution, with a and * for its double points. We have therefore this 
Theorem :— 
““ On each of the fifteen lines A,,,, the three pairs of derived points, of 
first and second constructions, namely the pair P,, the pair P., », and the pair 
Po, 3, compose an involution, one double point of which isthe given point Py; 
the other double point being the point P3,,, of third construction and first 
group, upon the line.” 
[104.] We have thus discussed the arrangements of the points P,, 
P, Pz, on each of the ten typical lines which connect not fewer than four, 
* That the two first of these three pairs belong to an inyolution, with those two double 
points, was seen in [101.]. 
