572 
volution of the rays, considered as paired with each other, or with them- 
selves: thus the second and third rays are the fouble rays of an involution 
(of the usual kind), in which the first is conjugate to the fourth, the 
fifth to the seventh, and the sixth to the eighth; while the first and 
fourth rays are the double rays of another involution, in which the 
second and third, the fifth and sixth, and the seventh and eighth are 
conjugate. 
[107.] It only remains to assign the arrangement of the two last 
points of second construction, P2,s, with respect to the other points, P,, Poy 
on a line A,,.2, or to some three of them; or to show how a*™ and a,* can 
be derived,* for example, from 8’, c', and a”: which derivation may — 
easily be effected, on the plan already described for the fifth and sixth 
typical traces. In fact, if we denote the six points a”c’s’a’/’a,*a™ by 
abcaBy, we have the three harmonic equations of [94.]; and if, by one 
of the modes of perspective, or projection, mentioned in | 95.], which an- 
swers to the initial arrangement [91.], we throw off the first point a’‘ to 
infinity, the finite line a*a,** is then quadrisected: being ztself bisected at 
a’, while c’ and B’ bisect its halves. In general, we shall have again 
the equations [94.]|, if we otherwise represent the six lately mentioned 
points on Bc’ by aByabe; and thus it is seen that those siz points are 
always homographic, in every state of the figure, or net, with the six 
points a/’B,%c™aB,c, on the fifth trace aa”, and with the six points 
ap “icp Bc,” on the sath trace p,a’’; in fact they are, if taken in a 
suitable order, the points in which the séa-rayed pencil [98.], with a’ 
for vertex, is cut by the line p/c’. 
[108.] We have thus shown for each of the twelve typical lines ['74.], 
in the plane anc, how all the points but three, upon that line, may be 
derived from those three by a system of harmonic equations, not necessa- 
rily employing any point P,, or other foreign} or merely auxiliary point: 
although it appeared that something was gained, in respect to elegance 
and clearness, by introducing, on the line aa’, such a point a* [99.]; or 
by considering generally, on any one of the fifteen lines A,,,, a point Ps, ; 
of third construction, belonging to what may perhaps deserve to be re- 
garded as a first group [1038.] of the points P;, in any future extension 
[1.] of the results of the present Paper. 
* This point ai? may also, by [81.], be determined on the seventh trace, or seventh 
typical line [74.], as the harmonic congugate of a’, with respect to Co and cy’. 
+ This non-requirement of foreign points is the only remarkable thing here : for the 
anharmonic function of every group of four collinear net-points is necessarily rational ; 
and whenever (abed) = any positive or negative quotient of whole numbers, it is always 
possible to deduce the fourth point dfrom the'three points a, b, ¢, by some system of auai- 
liary points, derived successively from them through some system of harmonic equations. 
