551 
fore 2475 in number; but as it is not difficult to prove that there are 
240 distinct cases of coincidence of line with plane (or of a plane con- 
taining a line), we must subtract this from the former number, and thus 
there remain only 2235 cases of intersection, of the kind which we have 
proposed to consider. Every one, however, of these 2235 cases, must be 
accounted for, either as a given point Po, or as a derwed point P, of first 
construction, or finally as one of those new points P,, of which we have 
proposed to accomplish the enwmeration, and to determine the natural 
groups, as represented by their respective types. 
[38.] We saw, in [27.], that each point Po, as for instance the point 
A, represents ¢welve intersections of the form A,° II,; and it is easy to 
prove that the same point Pp, represents twelve other intersections of the 
form A,-Il,,,; twelve, of the form A,,,°W,; and three, of the form 
As,,* T1,,,; but none of any other form. It represents therefore, on the 
whole, a system of 39 cntersections, included in the general form A+ 11; 
and we must, for this reason, subtract 195 (= 5 x 39) from 2235, leav- 
ing 2040 other cases of intersection of line with plane, to be accounted 
for by the old and new derwed points, P, and P,. 
[89.] An analysis of the same kind shows, that each of the ten points 
of first construction, as for example the typtcal point a’ [25.], represents 
_ one intersection of the form A,°TI,; six, of the form A, °TI,,,; scx, of the 
form A, * Tlz,2; st, of the form A,,,°11,; twelve, of the form A,,, - I1,,,; 
eighteen, of the form A,,,*TIz,2; etghteen, of the form A,,,° 11,; twenty- 
four of the form A2,.°TI,,,; and twenty-four others, of the remaining 
form A:,2*T1,,2. It represents, therefore, in all, 115 intersections A - 11; 
and there remain only 890 (= 2040 — 1150) cases of intersection to be 
accounted for, or represented, by the points Pp, of which we are in search. 
But all these 890 cases of intersection must be accounted for, by such 
new points, if the investigation is to be considered as complete. 
[40.] A first, but important, and well-known group of such points 
P,, consists of the ten points (already considered in Part I. of this Paper), 
INU ISAC IBS (CSE WEB USORG veziGl WAP 
namely, the harmonte conjugates of the ten points P,, with respect to the 
ten lines A,, which we shall call collectively the points, or the group, P2,; 
and among which we shall select the point 
a’ = (011), 
asa Third Typical Point of the Net. In fact, it is what we have called 
a point P,, because, without belonging to either of the two former 
STOUPS, Po, Py, it is an intersection A,*T1,,2; or rather, it represents six 
such intersections, of the line sc with planes of second construction, and 
of the second group: namely, with two such through B’c’, two through 
B,C,, and two through B,c,, being pairs of faces [28.] of three pyramids 
R,, inscribed in those three pyramids R,, which have been distinguished, 
in [26.], by the letters a,p, =. The same point a” is also the intersec- 
tion of the same line se with three planes I, ,; namely, with the three 
