D47 
of intersection, and sixty such cases conduct only to the five o/d (or given) 
points. This sort of arithmetical verification of the accuracy of an enu- 
meration of derived points, or lines, or planes, will be found useful in 
more complex cases, although it was not necessary here. 
[28.] Proceeding to a Second Construction [1.], we may begin by 
determining the lines A,, whereof each connects some two (at least) of the 
Sifteen points Po, P, but not any two of the five points Po, since otherwise 
it would be a line A,. Ifthe 15 points to be connected were indepen- 
dent, they would give, generally, by their binary combinations, 106 lines; 
but the zen collineations of construction, 
BoA, &C.; DAAz, &c.; EAA,, &c.; and EDD, 
show that 30 of these combinations are to be rejected, as giving only the 
ten old lines, The remaining number, 75, is still farther reduced by the 
consideration that we have (comp. (79)) the fifteen derived collineations, 
AA'D,, &C.; AB\C,, &C.; AC\Be, &c.3 Da‘Ay, &C.; EA'As, &e.; 
which represent only fifteen new lines, of a group which we shall denote 
by As, 1, but cownt (comp. [ 27. ]) as 45 binary combinations of the 15 points. 
There remain therefore only 30 such combinations to be considered; and 
these give in fact a second group, As,., consisting of thirty lines of second 
construction: namely, the thirty edges of the five new pyramids RB, 
C'B’AgA,, A'C’ByB,, BA'C2C}, AoBoC,D,, A B,C,D;, 
which are respectively cnseribed in the five former pyramids R, [ 26. ], 
and are homologous to them, the five given points a. . E being the respec- 
tive centres of homology ; for example, c/ = 4B‘ cpr, &c. The correspond- 
ing planes of homology will present themselves somewhat later, in con- 
nexion with the points P.. 
[29.]. On the whole, then, there are only forty-five distinct lines of 
second construction A,; and these naturally divide themselves into ¢wo 
groups, of 15 lines A,,,, and 80 lines A, », as above. Hach line of the first 
group Az,, connects one point Yr) with two points P, ; as each line A, had 
connected one point P, with wo points P); but no line of the second group 
Ao, , connects, at this stage of theconstruction, more than ¢wo points, which 
are both points P,. Through xo point ¥, therefore, can we draw any line 
A»,,; but through each point Pp we can draw three lines A:,,; and each of 
these is determined as the ¢ntersection of two planes II, through that 
point, or as crossing two opposite edges of that pyramid R,, which has not 
the point P, for a corner (comp. [26.]): for example, aa’D, is the inter- 
section of aBc, ADE, and crosses the lines Bc, DE. And besides being, as 
in [ 28.], the edges of certain other and inscribed pyramids z,, the 30 lines 
Ap,» are also the sides of ten new triangles 1,, namely, 
DjAyAy, &C.; C,B,A’, &e.3 CBA’, &c.; and a/z/c’, 
situated in the ten planes X1,, and inscribed in the ten old triangles t,, to 
