590 
[51.] We are now therefore enabled to assert that the proposed Hnu- 
meration of the Points Pv, of Second Construction, andthe proposed Classifi- 
cation of such Points in Groups, have both been completely effected. For 
the number of such groups P,,,, . . P23 has been seen to be eight, repre- 
sented by the 8 typzcal points, a/’..a™; which, along with the first given 
point a, and the first derived point a’, make up a system of ten types, as 
follows : 
a =(100); a’=(011); a” =(011); a’”= (211); a”=(211); 
a’=(021); a= (021); a= (121); a" ='(321); a™= (231); 
and the nwmber of the points vp, is (10 + 30 + 30 + 20 + 20 + 60 + 60 
+ 60 =) 290; so that, when combined with the points p,, they make 
up a system of exactly three hundred points, 2, Pz, derived from the five 
points Po. 
[52.] It is to be remembered that the three other ternary types, 
D, = (111), 49 = (111), a,” = (311), 
have been seen to represent points which are, in the guinary sense, syn- 
typical with a’, a”, a, and therefore belong to the same three groups, 
P,, P2,2, Ps,3; all these three points being in the plane azc, and on the 
line aa’p,. And it is evident that the five other points, 
Ay” = (012); a,7=.(012); a," (112); a," = (812); 4,5 = (218), 
belong (as has been seen) to the same five last groups, P2,4,.- P2,s, a8 the 
five points above selected as typical thereof, namely the points a*.. a™, 
and are situated on the same two typical lines, Bc and B/c’.. The tran- 
sition from a’ to B’, c’, or from a” to B”, co”, &c., is very easily made, by 
a rule already stated [20.]; and therefore it is unnecessary to write 
down here the symbols for these derived points, 8’, 8”, &c., or c’, c”, &e. 
But we must now proceed, in the remainder of this Paper, to investi- 
gate some of the chief Geometrical Relations which connect the points, 
lines, and planes of the Wet, so far as they have been hitherto deter- 
mined: namely, to the end of the Second Construction. 
Part I11.—Applications to the Net, continued: Enumeration and Clas- 
sification of the Collineations of the Fifty-Two Points in a Plane of 
First Construction. 
[53.] The plane azc has been seen to contain, besides the three points 
Py) Which determine it, four points P,, namely a’, B’, c’, and p,; and it 
contains forty-five points P,, namely the three points 4”, B”, c” of the group 
P2,,, and six points of each of the seven remaining groups of second con- 
struction. This plane II, eontains therefore fifty-two points Po, P,, Ps; 
and we propose to examine, in the first place, the various relations of col- 
linearity which connect these different points among themselves: intend- 
ing afterwards to investigate their principal harmonic and involutionary 
relations. 
