571 
and not more than seven of them; but there are still two other typical 
lines to be considered, belonging to the groups A, and A,,,; whereof 
one, as BC, passes through e7ght points [54.]; and the other, as v’c’, has 
ten points upon it [56.]. Beginning with the first, we easily find that 
the two sets of points, a’sc and 4/A,"a’, are triads in involution [85.] ; 
the latter set being thus deducible from the former: while the two other 
points upon the line may be determined by the condition that they 
satisfy this other involution of two triads, a’’sc, a’A\"A". With the tn- 
tial arrangement [91.], the line a“a," is trisected in B and c, and its mid- 
dle part sc is likewise trisected in a” and 4,"; while each line is bisected 
in a’, and cut at infinity in a”. And in general we may enunciate 
these two Theorems :— 
I. “ On every line of first construction, the point Pp, and the two points 
P, form a triad, the triad in involution with which consists of the point P,, 1, 
and the two points P., 4” 
TI. * On every such line A,, the triad formed by the point P,,,, and 
the two points Py, 2s in involution with a triad which consists of the point 
P, and the two points P., 5.” 
[105.] Besides these two involutions of triads, we have two distinct 
involutions of the ordinary kind, into each of which all the esght points 
enter; two being double points in each. For we have these two other 
Theorems, deducible, indeed, from the two former, but perhaps deserv- 
ing to be separately stated :— 
III. “ On every line of first construetion, the two given points are foce 
of an involution of six points, in which the points Py, Po,,, are one pair of 
conjugates, while the two other pairs are of the common form, Ps,4, Py me 
For example, a’, a” are such a pair, on the line zc. 
IV. “ On every such line A,, the points P,, Po,1, are the double points 
of a second involution of six points, obtained by patring the two points 
of cach of the three other groups.” 
[106.] Finally, as regards the remaining typical line B'c’, which con- 
nects two points P,, and passes through e7ght points P., if we reserve for 
a moment the consideration of the last pair, P,,s, or a* and 4,*, we have 
a system of eight points upon that line, homographic with the recent system 
of eight points on the line no; being indeed the intersections of the line 
ec’ with the eight-rayed pencil, 4.a’BoA/A,"AA UA", When taken in the 
order a/c’p/a/a,“#,%4,%4"4, No description of the arrangement of 
these latter points is therefore at this stage required : but as regards the 
pencil, it may be remarked that, by [104.], the Ist, 2nd, and 8rd rays 
form a triad of lines, in involution [85.] with the triad formed by the 
4th, 5th, and 6th; and that the ¢riad of the 2nd, 8rd, and 4th rays is, 
in the same new sense, in involution with the triad of the 7th, 8th, and 
1st: from which double involution of triads, the five last rays may be de- 
rived, if the three first are given. We have also by [105.] a double in- 
