552 
which connect, two by two, the three lines B’c’, B,c,, B,¢,, and contain 
the three points 4,p,. It is also, ins’z ways, the intersection of one or 
other of these three last lines A., 2, with a plane Ih ; in ¢hree ways, with 
a plane IL,,,; and in twelve ways, with a plane Il,,.; so that a single 
point P., , represents thirty intersections of the form A II; and the group 
of the ten such points represents 300 such intersections. We have there- 
fore only to account for 590 (= 890 — 300) intersections A II, by other 
QTOUpS Pr, 2, &C., of points of second construction. 
[41.] A second group, P,, 2, of such points P,, has already presented 
itself, in the case of the traces A, Bo, Co | 18. |, of the lenes A,Az, ByB2, 0,02, 
on the plane asc. The ternary symbol of the point ay has been found 
(77) (92) to be (111); its guénary symbol is therefore (11100), which is 
congruent (10) with (20011); hence in the full, or quinary sense [9.], 
this point Ay is syntypical with the following other point, in the same plane 
ABC 
a” = (211), 
which we shall call a Fourth Typical Point, and shall consider as repre- 
senting the growp P,.; this group consisting of thirty such pornts, 
namely of two on each of the 15 lines Ag, ;. 
[42.] Each of these thirty points P,, , represents seven intersections of 
line with plane; namely, two of each of the three forms, A,,1- Iz, 1, 
Ao, 1°12, 2, Az, 2°Tl2,1, and one of the form A2,2-M. For example, the 
typical point 4”, which is the intersection of the two lines aa'p, and B’c’, 
is at the same time the intersection of the former line A,,, with each of 
four planes II, which contain the latter line A,,,; being also the inter- 
section of this last line 8/c’ with a plane II,, namely apz, and with two 
planes II, , which contain the first lime aa’p, The group P., , represents 
therefore 210 intersections a-%; and there remain only 380 (= 590 
— 210) intersections of this standard form, to be accounted for by other 
groups of second construction, such as Pp, 3, &. 
[48.] In investigating such groups, we need only seek for typical 
pownts ; and because every such point is on a line of one of the three forms, 
A,, As, As, 2, We may confine ourselves to the three typical lines, 
BC, AA'D,, Bc’; or (Ofu), (tum), (ctw) ; 
in which, as before, « = €+ u, and in which the ratio of ¢ to wu is to be 
determined. And because a line in the plane azc intersects any other 
plane in the point in which it intersects the line which is the trace of the 
latter plane upon the former, we need only, for the present purpose, con- 
sider these lines, or traces: whereof there are, by what has been already 
seen, seven distinct ternary types, namely the following : 
[100], [011], [111], [111], [011], [211], [211]; 
which answer to the seven typical traces of planes, 
Wo At 
BC, AA/D,, BC’, A’B’C’, AA”, DA’, A/Cy. 
