553 
There are 22 (=3+ 3843414348 46) such lines, answering to 44 
3.243.343.441.24+3.14+3,.2+6.1) planes; namely to all 
the 45 planes Th, M2, except the particular plane azo, on which the traces 
are thus taken. And we have now to combine these seven types of lines, 
with the three symbols of points, -(Otu), (tww) (ctu), according to the ge- 
neral law, lz + my + nz = 0 (76). 
[44.] The line zc is itself one of the three traces of the first type; 
and it intersects the twelve other traces, of the five first types, only in 
points which have been already considered. The line aa’, is, in like 
manner, a trace of the second type; and it gives no new point, by its 
intersections with the eight other traces, of the three first types; but its 
intersection with the common trace 48/0", of the two planes a,B,¢, and 
A,B,C, [19.], which is the only line of the fourth type, gives what we 
shall call a Lufth Typical Point, namely the following: 
a’ = (211); or more fully, a = (21100) = (30011). 
This last quinary symbol shows that the point a” is syntypical with this 
other point in the plane axe, 
A,” = (31100) = (811); 
so that this plane contains six points P;, 3, which (in the gwmnary sense) 
belong to one common group, although their two ternary types are diffe- 
rent. In fact, the point a,’ is the common intersection of the line a/p, 
with the two planes [12111] and 11211], or B’c,c, and c’B,B,, as the point 
A” is the common intersection of the same line with the two planes 
[11121] and pak Gap OF A,B,C, and A,B,C2, as above. 
[45.] There are thirty distinct points P., 3, of this third group of se- 
cond construction ; and each represents two (but only two) intersections, 
which are both of the form A2,1°W2,2. The group therefore represents a 
system of 60 intersections A- 11; and there remain only 320 (=380-60) 
such intersections to be accounted for by other points, or groups, such as 
Po, 4, &c. It will be found that we have now exhausted all the points, 
or groups, of second construction, which are situated on lines A,, , ; but 
that two other groups of points Pp, may be determined on lines A,, by 
tee the typical line sc with the two last sets of traces [43. ] as 
ollows. 
[ 46. ] Combining thus zc with p,c” and D,B”, or with the traces [112 ] 
and | 121], we get the two following points, of a fourth group of second 
construction, 
AY = (021); a”, = (012); 
whereof the former may be taken as a Sixth Typical Point. , There are 
twenty points of this group P2, 4, whereof each represents three intersec- 
tions, of the form A,‘ IIz,,; for example, the typical point 4” is the com- 
mon intersection of the line nc with the three planes 0’A,A,, D,A,B,, D,AgBo; 
the group therefore represents sixty intersections A ‘1, and there remain 
260 (= 320 — 60) to be accounted for. - 
R. I. A. PROC.—VOL. VII. 4u 
