546 
may be called a First Typical Plane. As a verification, we see that 
when we make o =¢+u =0, in the second symbol [23.], and divide by 
t, we are led to the recent symbol for azc, as one of the planes which 
pass through the line zc. 
[25.] The derived points p,, of the same first construction, which are 
all, by [1.], of the form A,‘ Th, are in like manner ten ; namely the in- 
tersections, 
BC* ADE, &.; DA‘ BCE, &c.; HA‘ BCD, &c.; and DE° ABC, 
which have been denoted in [14.] and [15.] by the letters, or literal 
symbols, 
a’, &.; Ay, &c.; A,, &e.; and ,, 
and for which guinary symbols (49) (54) (60) (61) have been assigned. 
Of these ten points four, namely 4’, B’, c’, Dy, are situated in the plane azc, 
and have accordingly been represented [20.] by ternary symbols also : 
and we may take the particular symbol of this sort, 
a’ = (011), 
as a type of this group P,; understanding, however, that the full or qui- 
nary type is to be recovered from this ternary type, by restoring the two 
omitted zeros; so that we have, more fully, 
a’ = (01100) = (10011). 
And the point a’ itself may be considered as a Second Typical Point. 
[26.| We have thus denoted, by literal and by quinary symbols, 
whereof some have been abridged to ternary ones [17.], and have been 
also represented by types [9.], not only the five given points Py, but all the 
ten lines A,, ten planes I1,, and ten points P,, of what has been called, in 
[1.], the First Construction. And it is evident that we have, at this 
stage, ten triangles T,, namely the ten, 
ADE, &c.; BcE, &c.; Bop, &c.; and axe, 
whereof each is contained in a plane II,; and also five pyramids R,, each 
bounded by four of these triangles, namely the pyramids, 
BCDE, CADE, ABDE, ABCE, ABCD, 
which may be called the pyramids a, 8, c, D,E; each being marked by 
the literal symbol of that one of the five points Py, which is not a corner 
of the pyramid. 
[27.] It may be remarked, that ten arbitrary lines in space intersect, 
generally, ten arbitrary planes, in one hundred points; but that this 
number of intersections A, * II, is here reduced to fifteen, whereof only ten 
are new ; because each of the five points Py counts as twelve, since in each 
of those points four lines cut (each) three planes, while each of the ten 
planes contains three lines ; so that thirty binary combinations are not cases 
