Da7 
[58.] Each of these four last lines passes through six points; thus 
the trace [111] passes through the points (011) (101) (110) (211) 
(121) (112), or through 4” B” c” a B™ oc”; [011] through (100) (011) 
111) (111) (211) (211), or 4a” By cp) c™ B,™ ; [211] through (111 
g 
(011) (102) (120) (213) (231), or p, a” BY ¢,” c™ B™; and [211] 
through (011) (111) (131) (120) (102) (281), or a’ cB,” ©,” BY a™; 
the correctness of the ternary symbols being evident on inspection, if the 
law lx + my + nz =0 (76) be remembered: and the literal symbols being 
thence at once deduced, by [51.] and [52. ]. 
[59.] So far, then, that is when we attend only to the twenty-two 
traces [ 43. | of planes II,, II, on the plane azc, we find a system of three 
collineations of eight points; three of seven points; three of ten points ; 
and thirteen of six points each. Each collineation of the first of these 
four systems counts as 28 binary combinations of the 52 points in the 
plane [53.]; each of the second system counts as 21 such combinations ; 
each of the third system as 45; and each of the fourth as 15. We there- 
fore account, in this way, for 844+ 634 135 + 195 = 477 binary combi- 
nations; but the total number is 26.51 = 1326; there remain then 849 
to be accounted for, by lines A; which are not traces, of any one of the 
foregoing groups. 
[60.] In seeking for such new lines, it is natural to consider first 
those which pass through one or other of the three given points A, B, c; 
and the types of such are found to be the five following, each represent- 
ing a new group of six lines A;: 
[021]; [021]; [031]; [032]; [031}. 
As symbols, these answer respectively to the five new ines : 
(100) (112) (012) (112) (312), rae ay 
(100) (112) (012) (112) (312), or ac” a,” B™™ av: 
(100) (118) (218), or ac,” 0"; 
(100) (123) (128), ‘or ac,™ Be; 
(100) (213), “or AA. 
We have thus twelve lines A, each connecting a point P), with four 
points P,, and counting as tex binary combinations; twelve other lines, 
each connecting a point P) with two points P,, and counting as three 
such combinations; and se lines, each of which connects a point 
Po With one point Ps, and counts as only one combination. In this man- 
ner, then, we account for 120 + 36 + 6 = 162, out of the 849 which had 
remained in [59.]; but there still remain 687 combinations to be ac- 
counted for, by new lines of third construction, which pass through! no 
given point. 
