550 
get o = 0, or [11110], or Ex‘B,c’c,, again as in (95), if we oblige it to 
contain E, or B,, or c,. But there remain the two points a, and A,, de- 
termining the two new planes B’c'a, and x’c’a,, for the former of which 
we have ¢+o0=0, or u=- 2t, c =—¢, and therefore have the symbol 
[11121]; while for the latter we have u = ¢, « = 2¢, and therefore the 
syntypical symbol [11112]. There are twenty planes of this group T1,,., 
as may be at once concluded from inspection of the type ; among which 
(comp. [19.]) we shall select the following, 
plane ,B,¢, = [11121], 
and call this a Third Typical Plane. And it is evident that these 20 
planes II,,, are the twenty faces of the five inscribed pyramids BR, {28.], 
of which the edges have been seen to be the thirty lines Az,» On the 
whole, then, there are only thirty-five planes I], of second construction ; 
which thus divide themselves into two groups, of fifteen and twenty, re- 
spectively. 
[35.] To verify arithmetically (comp. [27.] [28.]) the completeness 
of the foregoing enumeration of the planes 11,, we may proceed as fol- 
lows. In general, fifteen independent points would determine 455 planes, 
by their ternary combinations; but the 25 collineations [28.], which 
give only the lines A, As,;, account for 25 such combinations, leaving 
only 430 to be accounted for, by so many triangles. Now each plane 
II, contains three points P,, and four points »P,, connected by six colli- 
neations; it contains therefore 29 (= 35 — 6) triangles, and thus the ten 
planes IT, account for 290 triangles, leaving only 140, situated in planes 
II,. But each of the 15 planes 1,,, contains one point P), and four 
points P,, connected by two collineations; it contains therefore 8 
(= 10 — 2) triangles, and thus 120 are accounted for, leaving only 20 ter- 
nary combinations to be represented, by triangles in ot her planes My. 
And these accordingly have presented themselves, as the twenty faces 
Ilz,> of the five inscribed pyramids R,. It must be mentioned, that the 
enumeration and classification of the foregoing lines and planes had been 
completely performed by Mozrvs, although with an entirely different 
notation and analysis. 
[86.] Itis much more difficult, however, or at least without the aid 
of types it would be so, to enumerate and classify what we have called 
in [1.] the Poznts p, of Second Construction; and to assign their chief 
geometrical relations, to each other, and to the five given and ten (for- 
merly) derived points, Pp and P,. In fact, it is obvious that these new 
points P, being (by their definition) all the itersections of lines A, or 
A, with planes I, or I1,, which have not already occurred, as points Py 
or P,, may be expected to be (comp. [2.]) considerably more numerous, 
than either the lines or the planes themselves. 
{37.] The total number of derived lines and planes, so far, is exactly 
one hundred ; namely, 55 lines A, and 45 planes HH, of first and second 
constructions. Their binary combinations, of the form ATI, are there- 
